I am quite certain that there's no easy way for governments to raise TFR. Hungary and S.Korea have tried, with relatively generous grants and incentives for parents, but the results are at best equivocal in Hungary's case and null in S.Korea's. What they're doing clearly isn't working.
So there are two options left on the table, which are bad but not utterly unreasonable:
1. Huge cash payouts for parents of children coupled with extremely generous incentives, both far beyond anything offered today. e.g., a $100k to $200k grant per child upon birth, with all hospital fees associated with childbirth covered by the state, free access to government childcare, and extremely generous tax credits for parents.
This might result in the wrong "sort of people" giving birth to many children... but at this point it's the one thing that hasn't been tried, it would resolve the concerns you mention in your post in a totally general sense, and it could also be a way to raise underclasses from poverty.
2. Manage the decline by investing in industries that can support an aging population, in programs that aim to pull people from decaying small towns into regional hubs, and by renormalizing retirement and pension schemes.
If there's a third way that isn't crazy -- which is to say bad _and_ unreasonable -- I'd like to hear it. (Examples of crazy ideas might include buying newborns from parents and raising them in government institutions, breeding children in artificial wombs, totally unfettered mass-immigration schemes, etc.)