There are some issues with your premise.
1. It's not a pure scientocracy
While there are not enough resources to support everyone, the most logical thing to do is use the remaining resources for those who are best suited for rebuilding the state of the scientocracy.
What you're describing is an autocracy. The governing's body prime intention is to perpetuate itself; and prioritizes itself over its citizens. In essence, the governing body becomes self-sustaining (providing future power for itself), as opposed to deriving power from their citizens.
Note that this isn't impossible. A governing body can be both a scientocracy and an autocracy. Most governments consist of several ideologies. The issue is usually more focused on the prioritization of these ideologies.
To use a real world example: Even though the US is democratic (the governing ideology, not the political party), it uses some forms of oligarchy (e.g. the President appointing Supreme Court judges, as opposed to holding elections).
Don't look at your government as an archetypical example of a scientocracy. Consider them as a mixed-ideology-government that predominantly operates as a scientocracy, but also relies on fall-back ideologies.
E.g. in cases where science cannot help (e.g. morality), or in cases where there is no conclusive scientific evidence (yet); how does your government approach the topic? Does it err on the side of maintaining order? Does it err on the side of personal freedom?
Note that if you do limit yourself to only scientocracy (above all else), then the solution is simple. The argument against eugenics is a moral argument. Objectively speaking, eugenics will improve on the human condition considerably, in a way that humanity gets to decide what to improve.
If you are looking at things purely scientifically, and by extension objectively, and put the objective above the subjective at all costs, then the moral argument is moot and eugenics will be implemented on the basis of its objectively superior results.
2. This is not eugenics.
Frederick Osborn's 1937 journal article "Development of a Eugenic Philosophy" framed it as a social philosophy—that is, a philosophy with implications for social order. That definition is not universally accepted. Osborn advocated for higher rates of sexual reproduction among people with desired traits (positive eugenics), or reduced rates of sexual reproduction and sterilization of people with less-desired or undesired traits (negative eugenics).
Eugenics revolves around genetic selection. While this can be artificial (e.g. gene mapping), the initial approach was to artificially enhance the "natural" genetic modification process, by matching partners more intelligently (as opposed to ineffable feelings of love or lust).
Not having enough food to feed everyone, and choosing who to feed, is not eugenics.
Note that it does touch on the same evil: deciding who lives and who dies; and therefore inherently "rating" the value of human life.
There is overlap between the two, but this is not eugenics. And this is important! Opponents of the new system will call it out as eugenics (and thus a bad plan). In response, those who advocate this system will spend a lot of effort pointing out that this is not eugenics.
This is already a first answer to your question: if the advocates of this system sway more people than the opponents do, then it will gain popular support.
3. Putting the cart before the horse
Question: It's tempting to turn the moral cheek and go for liberal policies in such a crisis, but the scientocracy is dead-set on eugenics. How does the scientocracy proceed -- what are logical next steps for it to take? What heuristics should it base its next actions on?
I don't understand your question. When you say that "the scientocracy is dead set on eugenics", then you've already reached the point where the people in power are willing to implement eugenics.
By definition of them being the people in power, they are obviously capable of implementing such a system, so what's stopping them from simply implementing it?
- If, in a dictatorship, the dictator wants something to be done, then it will (should) be done.
- If, in an oligarchy, the oligarchs want something to be done, then it will (should) be done.
- If, in a democracy, the majority of voters wants something to be done, then it will (should) be done.
- Logically, if, in a scientocracy, the empirical evidence proves that eugenics are an objectively superior approach, then it will (should) be done.
This is almost exactly the definition of what a scientocracy is: the course of action is decided by the empirical evidence, not by people's subjective opinions.
I suggest you take a step back here. Assume that there is no (conclusive) empirical evidence yet. But the current evidence strongly favors eugenics as a superior approach.
Because it's unproven, neither side of the debate can prove that they have the answer. And because they can't prove anything, the discussion devolves into assumptions and beliefs.
Because even if your citizens are all logical and willing to await conclusive evidence; they will still need to deal with everyday matters until that point. And in absence of conclusive evidence, they will have to go by their gut. Especially if they are highly logical, they will be very inexperienced with making gut calls, and there will be a higher chance of them making the wrong gut call.
This imprecision in the absence of conclusive evidence, is exactly what can cause people to make wrong decisions (even with the best of intentions).
But how do we implement this food selection system?
Okay, so we've addressed the inconsistencies. On to an actual answer. Note that while I still believe that your example is not eugenics; I'm going to consider that the "food selection program" is equally abhorrent to the public as eugenics is.
This is mostly because the opponents of the "food selection system" will call it eugenics in order to discredit it, and therefore those who are against it (the only people you need to worry about) will consider it equally evil as they do eugenics.
Oh boy, it's time for my favorite quote!
Those who play with the devil's toys, will be brought by degrees to wield his sword.
In other words, it starts out innocent (toys), but it expands step by step (by degrees) until it's no longer innocent (sword).
There's also a runner up quote to use here: Divide et impera (divide and conquer). You'll see why.
This is almost always the way in which controversial topics are introduced to the public. A quick and sudden change will cause a massive outcry (i.e. why revolutions are generally bloody affairs), but a slow and steady change may go by unnoticed.
The approach needs to be tailored to the actual goals. So let's draft a first example of the intended food selection program, in ranked order of priority:
- Those who contribute to the government are given food.
- Those who meaningfully contribute to scientific research are given food.
- Those above certain IQ are given food.
- Those who meaningfully contribute labor (engineering, social facilities) are given food.
- The rest of the food can be distributed among the population as they please (however the people choose to distribute it, doesn't matter as long as they don't start a civil war).
If you were to implement all of these rules at the same time, that's tantamount to a revolution; which will not go over easily.
So instead, let's play with the devil's toys:
1. Set a precedent for denying food.
Focus on a group of people which is widely considered as evil/no good. Using a modern day example: ISIS.
- Gain the public's approval to prohibit (or selectively limit) the food supply to regions that are under ISIS control.
- Try to offer (and attempt) several other solutions first, before mentioning the food. That way, it doesn't come across as if it's your main goal.
- Ensure that the publically known intention is to eradicate ISIS. Do not mention the food selection program.
- The true end goal is to get the people to agree to denying food to ISIS, because it sets the precedent that food is no longer an inalienable human right.
- If you want to improve the results unethically: secretly fund ISIS, to make them stronger. The stronger ISIS is (or appears to be), the more likely the people are to agree with your harsh food control measures. However, this does come at the risk of your plan being exposed.
2. Show examples of people who should be given food.
Focus on a person/group who is widely considered as good. Using a modern day setting: a researcher who is close to finding to cure for cancer, but who is unable to continue his research due to local rioting due to food shortage.
- Do not suggest that the food supply needs to be increased.
- Instead, merely report on the problems with continuing the research. Let the media explain to the people that riots are the cause of the problem.
- Increase public interest in the problem (e.g. by calling it into focus repeatedly)
- Eventually, the media will inform the people that the rioters are rioting over a food shortage.
- It's also possible that the rioters themselves come forward, explicitly pointing at the food resource problem (similar to how Al-Qaeda publically denounced America in its released videos)
- To speed things up, have someone unconnected to your government suggest that the problem can be solved by ensuring access to food.
The core idea is to let the people come up with the idea to send food. This creates a precedent that food can be given to those who meaningfully contribute to science.
3. Discredit the groups that won't get any food from the food selection system.
Even though the food selection system is not used yet; you'll already be fairly certain that certain groups will get the short end of the eventual stick. For the sake of example, let's say that you know that these groups will likely get no access to food: prisoners, janitors.
The goal is to dehumanize them. Don't mention whether they deserve food or not, but simply try to lower their perceived status in society.
- Prisoners: Provide exposés on the horrible consequences of crime. Highlight innocent casualties. Paint criminals as wilfully criminal (and not by necessity). Become highly intolerant of lesser crimes, edging towards zero-tolerance. Try to foster a public opinion that casts criminals and ex-convicts as outlaws, people who society no longer includes and who will have to fend for themselves.
- Janitors: Focus on news stories janitors who commit a crime or do something that the public disapproves of. E.g. find a janitor who deals drugs in the high school they work in.
- At the same time, increase public opinion of roombas and automated cleaning facilities. This drives the point home that janitors are becoming redundant in today's society.
The goal is to lower the societal rank of these people, and foster public discontent about them.
You don't need to even raise the issue of food scarcity. We've already established that people who are deemed "unworthy" by the public (e.g. ISIS) should not get a food supply.
All you need to do now is ruin the reputation of criminals and janitors, up to a point where people will start equating them to ISIS, and therefore will treat them the same way.
And your hands are clean, since you never suggested to stop feeding prisoners or janitors.
4. Indirectly promote the added value of the government to society.
This can go hand in hand with the above points, but can also be approach independently:
- Pull attention to issues that the government expertly handles. The outbreak of malaria was handled perfectly? Pull it into focus. Repeatedly find out why the system handled it so perfectly (spoiler alert: the answer will always be that the benefits are inherent to government).
- When publically condemning janitors and prisoners, make sure that the people see the government as a system that keeps them safe.
- Always keep focus on would-be-problems that would occur if the government was disbanded. E.g. one of the main reasons why Kim-Jong-Un stays in power, is because his people believe that the NK government is the only thing that prevents a hostile invasion by America/the West.
- Cover up the bad things in the government (e.g. corrupt officials). Excessively highlight the good things.
Some quickfire suggestions (too small for their own topic):
- Evolve towards giving food as prizes in contests.
- Put a high stress on fine cuisine.
- If you (freely) give food to a subset of the people (e.g. the homeless), that will have a positive impact on a larger subset of the people (thus netting you more bang for your buck).
- Never raise the topic of food rationing for publically divisive topics. Only open that door once you know you have public support.
- As you've noticed, a lot of these steps require a decent media infrastructure. Closely control the media (but remain absent enough to not be exposed), ensure that everyone is incentivized to listen to the media.
5. Rinse and repeat
By using the four steps listed above, you've created four (independent) public precedents:
- We should curtail the food supply of the unworthy.
- We should ensure the food supply of the worthy.
- Some groups are not meaningfully contributing to society.
- The government is essential to prevent the system from collapsing.
And these are the cornerstone of your food selection program.
The people may not immediately switch over to this new ideology. E.g. even if they agree to curtail the food supply of ISIS, they might not agree with the methods on principle.
Which is why you need to repeat the steps. Find another enemy whose food the public will happily take away. Find another public hero who the people willingly give food to. Ostracize more groups (keep it unrelated to food rationing). Continually improve the public image of your government.
Eventually, the people will put two and two together, and they will start living with an ideology that is more compatible with your food selection program.
And the best part is that they think it's their own decision. They don't feel like the government is pushing them towards food rationing, they are simply deciding to use food rationing as a punitive measure, and they chose to do so voluntarily.
As a footnote, I hope you agree that your scientocracy is the narrative evil of your story. Because my answer is basically adapting the fascist playbook to your situation.