Make yours better first, eliminate other later
Your first goal is to build enough GCs (Gowerment Clinic) to be able produce the right quantities of childs. (If you want decrease population, you will do it later. Now you want to be able satisfy all requests - or at least nearly all)
Those childs would be genetically enhanced in the process, to be more successful (inteligent, pretty and more resistent to illness) - if you are starting this technology, there would be not much enhancement first, but as it will widespread, it would became better. This would them make more attractive to parents, than natural one (no problems with pregnancy, better results in school and life, less hurden with child illness - so better life quality for parents).
You will start to offer it as a priviledge to rich, bussy, celebrities etc. and actively promote it in press - less "she is pregnant" titles, more "She get the perfect child" and "The child is soo much wonderfull". Also hire someon from Apple PR section for the campain :) Not all would opt in, but many would and your capacities at first are limited anyway. (On the dark side you secretly support movement "Why only rich can get all the best, we want too".) Once you can mass produce childs, you go down with the price (at public request ofcourse) and even subvent it, so the middle class could affort it easier. (You need more requests, than you are able to fullfill just now, to be able pay for increasing production capacity and justify it - you still support the first movement too. You start also opposing movement "It is unnatural" and put the right figures in it - such, that would be easy denounce and keep huge stream of scandals about them - to discreditate the "unnatural" idea with its protapagators. And you also secretly support antivaxers. Also make some vaccine scandals and as consequence request more testing, more burocracy and more effort to prove, that new vaccine is really-really safe.)
At the start simple media manipulation, curiosity and beneficity of results would create the demand. Once you are near to be able supply enought to satisfy the natural demand, you need to go futher and decrease the prices again (yes, you are allready selling below the cost, but you want to offer it cheap and later totally free). Some funds could be drawn from common medical support (well GC are medical too, just redirect more funds their way). Let independent sources notice the fact, that the new childs are more healthy and less prone to illness (not totally imune, but less prone and have lighter symptoms in case of sickness - support such reports, discourage opposite one).
This can be reached in one or two decades and many of your childs are in school already, much more growing up to this age - they became seen as "normal minority", not just something "experimental". Even poor families can now affort to have one and there will be some allowances to rise one (under many exuses as "medical research data" and "right child care" and "propagation of better health for everyone (well it goes just to those, who have it engeneered from birth so guaranted)" ) (and you would tolerate abuse of the system, because you want to have as much of such childs as possible - even in you ideal world would be need for garbage-collectors).
With growing antivax movement, lower clasic medicine funds (and maybe even with little dark support) more normal illness would spread, and also their little more agressive variants. Superiority of your childs is visible and "common knowledge" (well even when not "vaxed", they was getting some "immunisation" while was created from the start and as part of the process), while widespread of disseases is also nothing new. Your milestone is about 1/10 of new population in developed countries came from GCs (or some similar sources - you cannot totally suppress concurecy now, but at least they have to stand to some "quality standards", including full immunisation. And well your prices are really dumping, so only rich would choise alternative source). Even in undeveloped countries there are GSs and your childs are at least tolerated.
When first vawe of your childs is mature and good portion of them is successful in their jobs and planing to settle and create families, some pandemy occurs. Say like COVID-19. Youre childs are good, usually does not affected at all, the rest have just little problems. Their position in society is hardened and they will opt for GCs childs more, then for "natural". Yes, even their natural childs would be better than average, but it is not problem now, you are good with it - even you push it in media as much as possible. At this scale it is clear, that your "new race" is better, but still is everything fully optional and now everybody have a choise - natural or GC - and can get what he/she want. Finally pandemy is over, your "race" is still minority, but really successfull minority and for more and more people is going to give their offspring the best they can - born in GC and be one of those successfull new people. As usual their are conspiration theories everywhere (as they are in Real World today), but it is normal. And well, some of new childs was affected too, after all. Some even died. (Well mostly those with unhealty life and preconditions, but propaganda does not care - they died after all.)
Normal desseases are more and more common now (antivaxers still get their support and new vaccines are stil harder and harder to develope and take longer - all for "public safety"). GSs are still delivering for free (or just nominal price) and are everywhere. The population is growng much more, than before - as you produce your growing share, but natural born is still common too. Time to reduce it a little. Time
to next pandemy, just little more lethal - lets 1/10 of total population die. Mostly those normal born and so even more problem for developing countries and religionar people, who are natural-born strict. It is bad, but world will goes as usual, 9 from 10 people survived (or 7 from 10 in some localities), so it could go the normal way. And we have medially survived second pandemy. Yay!
Now is time for even more media push on how the GC born did a more better, then average population. Get more customers, get more share and prepare for future. Normal dissesases gets their toll on natural-borns, it is something normal. Now more than 1 of 4 newborns are GC originated. For next decade we finally came with our secret card. This time the conspiracies will became true, but as it was proven false before, nearly nobody would trust it. We really have (be it from start or developed in last 60 years with growing GC born population) artifical virus designed to be mostly lethal for those without specific gene sequence and nonlethal (but still infectious) for those, who have it. And it would also hit at fartility of anyone infected. Not fatally yet, but hard.
As the normal status is to have a lot disseases around and sometime pandemy, it is just bad case this time. And all are hit on their fertility, even GC borns, so it is harder to multiply naturally and more work for GCs to fill the missing population. In next generation GC borns will be majority and you are big step nearer your goal. And with a lot and constant decimation of population, it would slowly change to work in such situation too. Couple of decades is long enought time for that (compare life now and before WW 2).
Your GC borns are able to carry all the work somehow, at the worst, but there is still a lot of natural born around. Also many are fertile (both Nb and GCb), but now it is time to reflect the situation and slowly modify the law to more and more propagate GCb - for critical positions it would be "high recomended" to hire GCb as there is better chance, that next pandemy would not paralyze the life as much as previos three (or four). The natural born is now officially presented as worse choise (as smokig, or drinking for example) and so some penalties are applied, some allowances are withdraw and each iteration it is more and more restricted.
And nobody yet realised, that from some point all GCb are created infertile - they are too young yet, fertility was affected by many desseses and mainly by last pandemy and GCb usually opts for GC anyway.
You are nearly there. You can build-in new sequencies in GC for next generation, you can select on that sequence in pandemy later and erase all those fertile yet from the first "not-so sofisticated" GC borns from times, when infertility may be seen as "issue", (and erase all natural born too - if they are not in constant quarantine, their are dead, if they are in, they do not matter anyway - they not have resources to multiply adequatly).
With regular updates to genetic sequencies you get not only better workers (at your choise mor inteligent, obedient, whatever) but also marks to target, if you need erase older models.
And for start you need "just":
And 100-300 years of time.