answering as prompted in your other question...
To be honest, assuming your starting world is anything like our world, getting anywhere near 50% take-up sounds beyond fiction.
Going through your suggested reasons:
Attractiveness – most of us – even if we aren't particularly attractive – are happy, and it's likely to look like some furry fetish if we transform into flawless animal-people. Also, cosmetic surgery already exists, but most people don't take it up not due to price, but because they don't want it.
Healthy body – some might take it for this reason, but a quick google suggests only a very small proportion of the (non elderly) population has a life-limiting physical disorder.
Forget trans; even optimistic estimates suggest only 1% has "some degree" of non-binary conformity. It's not going to give you a significant proportion. Additionally, I'd imagine someone who identifies as a female human isn't going to want to be a female cat any more than a male human
Prefer the species – I think many of us might wonder what it's like, but not want to do it long-term. The grass is always greener on the other side. I imagine a lot of people might take the 1 week trial each time, then revert.
Initial Offerings
For the first offering, anyone even vaguely sensible would avoid an offer of a species change. There's too many risks & unknowns. Will it work? Are there side-effects? Are they really killing people and replacing them with beastman puppets? What happens to my memories & personality?
There's enough people who avoid the first release of an OS because it'll be problematic, how much more an unproven supernatural species change! A 1 week trial is also worth nothing on the first offering, as there's no proof yet that you can change me back!
So once you've had at least a couple of offerings, and people are starting to believe it's for real, and doesn't have any major immediate side-effects...
It's likely that the first people who take it up will be people with a furry fetish, or sex-workers who want to / are forced to monetise that. So that immediately means that – unless furry fetish has suddenly become respectable & desired (it won't in that time – look how long it's taken LGBT to become acceptable) – most people are going to look at it in a somewhat dim view.
I think this is worth considering, because those who first embrace it are likely to set the tone for the overall thing.
Breaking it down
Let's deal with some easy demographic blocks...
I'm going to skip 'Christian' (31%) because it's a very varied block.
Wikipedia gives ~24% of the world's population as Islamic, and most of that will be in countries with some degree of Islamic law. Taking Indonesia as an example (it's one of the most liberal Islamic majority nations, and one I'm somewhat familiar with), the culture is still largely conservative, and species shifting is very likely to be seen as Haram. This is doubly so if species shifting is seen as a 'western' thing – which is likely will as the countries most likely to give it good press are western – or a fetish. So religious observance and social pressure will prevent them from taking up the offer – and in many Islamic nations, there would be punishments (either official or done by local communities) for anyone who did accept the offer.
China (18%) will be your next largest block. It seems unlikely that the Party would look favourably on species shifting. It would likely be seen to go against the One China value. The current party is more traditionally communist, and has been stamping out anything seen as foreign / disruptive (e.g. clampdown on Christianity in China), so I think we can safely assume they would similarly take a very low view of species-shifting. It would also possibly be seen as disrespectful to one's family / community. It would be a career-limiting move.
Your next biggest demographic block is Hindus (15%). My knowledge here is rather limited, so perhaps a commenter who knows more might help. For lack of anywhere better to start, comparing attitudes to LGBT issues, Hinduism is more accepting than Islam, but it's still not as accepted as in the west. Hinduism does of course have a big connection to anthropomorphic animals in its gods – I suspect that would make it reject shape-shifting, as e.g. it would be seen as disrespectful to Ganesha to accept an elephant trunk. On the other hand, it might mean a larger uptake. However, again, very strong family pressures are likely to prevent people taking up the offer.
We've now dealt with 57% of the world's population... You'd now need near 100% uptake in the rest of the world to reach your goal.
Let's go to the other extreme – Western Europe and North America (~14%)
Views here are generally very liberal.
I think we can assume that uptake would be largest amongst the young (mid-teens to mid 20s). Younger children hopefully aren't permitted to make life-altering decisions like species-shifting for themselves? Also once you move up the age brackets, views tend to be more conservative. So let's assume it's highest amongst 15-35s (~25% of UK population, probably similar across WE/NA).
Initially, it's going to be a career-limiting move. Even if governments move to make discrimination based on species illegal, discrimination based on sex has been for decades, but most companies still have a huge gender pay gap. It's going to be hard to be taken seriously in a executive meeting with rabbit ears. Based on other discrimination issues, it's likely to take decades for it to become widely accepted.
It'll also be problematic practically – in particular, any changes like tails or wings, or changes to basic skeletal structure (animal legs) will be hugely problmatic. As noted in your other thread, it'll cause issues for clothing. But also chairs. And cars – I won't want a tail if it stops me driving (or makes it difficult / uncomfortable). Modifications to cars – because of safety certifications – will take years. New furniture will be expensive. Similarly medicine – these new bodies might start healthy, but presumably they'll decay like ours. Medicine will take ages to catch up. Will I have to visit a vet to get my tail / wings looked at? Why is my fur/feathers dropping? I've got lice. Vets aren't licensed to work on humans in most jurisdictions.
There's many jobs where it'll be impractical – a safety hat won't fit big ears; a tail will be a liability in a kitchen. Whilst some might give benefits (wings for couriers...?) the majority of tools and workplaces aren't ready to take advantage of it. Paws won't work a mouse and keyboard. And flight is likely to get legislation pretty quickly (think drone laws), due to risks (dropping your bag at 100ft and killing someone), and privacy fears. There might be compulsory registration – knife laws are pretty pointless if gangs have retractable claws. Every nation will consider how it can be militarised.
With the increase of 'far-right' groups, one can quickly imagine "Keep Europe for Humans" marches (EDL, etc.). Short of actual lynchings, this will create social pressures against species-shifting.
Let's optimistically say uptake in WE/NA, amongst 15-35s (where it'd be highest) is 10%. Let's say 5% across WE/NA. That gives you probably less than 1% worldwide uptake. More realistically, it's likely to be an order of magnitude lower for many decades, without some other pressure.
So... what could you do to encourage people to change?
To make people want to change in large numbers, you'll probably have to invent some big issue, like a disease epidemic which the new species (and perhaps some humans if you want to keep them) are immune to. Or large cash payments, or real-estate in the beastman world across the portal, etc. Or a food shortage – perhaps the new bodies are able to digest grass & leaves etc.?
Sorry... that's all really negative! You might want to just skip your story forward a few decades to the point where uptake is now ~10% and starting to spike up? Or just set it somewhere like California or Vegas where uptake might be highest? Or start with a future world with widespread cybernetics & surgical bodymods which would be much more accepting of this offer.
