You send people off to war, but this is misaligned with their survival incentives (to stay home and have kids). This applies individually - but also culturally to how they raise their own children. Even military parents sometimes try to dissuade their kids from following in their footsteps.
A society pre-commits to supporting fertility treatments to raise children for fallen soldiers, so that depending on their behavior, even after death their family would continue to grow. This breaks the local "I have to survive" link. Similar to the promise of rewards in heaven - but in reality, implement strong tracking for rewards.
We are likely to observe self-sustaining phenomena: viruses, MLMs, cults. We are not likely to see things which don't grow or adapt. Or, we may evolve into something like this - we already have generation-level social tracking which correlates with reproduction (money, status, fame). The lack of morality of a system like this doesn't protect us from it.
Or, traits a society like this might need to have to actually work.
In poker you try to play to maximize your total expected value over many hands (many lives). But in life you are strongly punished for "losing" a hand, and sometimes people don't play to maximize the expected value to humanity because they are individually tied to their "hand" (life). In poker, a single entity takes all the wins and losses from all hands. But in life we individually treat our score as just the result of our own hand. Some people do expand their view to include the score of their family, region, culture, or nation.
We observe many successful memeplexes like this, sometimes grown out of control and very individually damaging
In the same way that playing many hands of poker makes it easier to play well (since you aren't tempted to over-value any single situation), allowing individuals to maximize expected value while disregarding individual survival can be net profitable. Not only is this related to the idea here, but it also applies to group membership in general. Feelings of being in allied group (military, tribe) can free people to have group-level optimal behavior rather than individual.
Were old Levirate (women marry the brother of their widow) practices a primitive version of this? Beyond the obvious form of serving as a pension/backup, this also has a component where even after death you have additional children you are somewhat related to (1/4 rather than 1/2).
Pensions are another similar idea.
Whose correlation with stable strategies & our desires is variable.
At the end of someone's life, evaluate how they did - their treatment of others, achievements, etc. Then decide how many descendants they should have. This could be a terrifying society, and easily exploitable by a cabal of old men. Arranged marriage societies in some ways could be viewed as this - but there's a limit to their power since they can't transcend death. Did such societies devolve into something exploitative? Would allowing them to individually reward men or women with extra children, after death, on net improve behavior?
If someone is judged as bad at the end of their life, kill them and all their descendants unto the 7th generation. i.e. in the bible or recently with Saddam Hussein. Obviously super unfair and anti-individualistic, nevertheless the threat of such punishment would force people to think more systematically.
Separating conception of a child from raising of the child means the family influence aspects of child-rearing are lost. There is doubt about how much this matters, but it's probably nonzero. There are technological adaptations to help here; detailed video diaries & recordings. More group living styles reduce extreme dependence on just the child-parent relationship; given that in such a society, a lot would be known about relationships, it would be possible to cause connections to be formed between generations. Again, tribal living emphasized kin much more than WEIRD people do.
It has some advantages in being able to concentrate rewards more intensely. How much of our modern wealth is explicitly because of inequality which riches allow? Extreme rewards lead to extreme behavior, in this case to our net social benefit.
But it's risky to concentrate the right to decide who has kids in any single group. The relative legal freedom we have around reproduction is a great long-term protection - compared to how regulated we are in everything (bike helmets, food warnings), it's amazing how such a major thing (creating a person and raising them with no training) is nearly totally unregulated. This is kind of good - no matter what crazy beliefs people have, in the worst case if they become super unadaptive, resistant individuals will at least be able to grow their way around them. Anti-natalism is locally self-defeating, and resistance to social control of reproduction makes it hard for system-level anti-natalistic beliefs to spread on net.
In fact such a drastic change may not be needed to handle falling TFR. Subcommunities which find a way not to suffer from this are already growing, just like always. Yet again averages are deceiving - TFR 1.5 doesn't mean we are destined to die out; it breaks down into subgroups with TFS <2.1 and >2.1; one of which is dying out and one of which is growing.