I don't see how the problem would arise. Genes replicate by increasing the number of people. Memes (of the kind we'd like) replicate by infecting existing people. So by diverting human efforts from genes to this kind of memes, we reduce population growth.

Yes, we (might) reduce the growth of the human population. But then we get memes replicating on human substrate, which means that memes have a Malthusian problem—they reach replicate to subsistence level of “resources”, i.e. humans.

The second order problem is that memes that cause their substrate to reproduce more and pass the meme to children have a reproductive advantage for meme natural selection. Which, if big enough, can lead to a parallel Malthusian problem in both human and meme populations.

When non-DNA inheritance gets better at using the available resources, a likely result is more effective competition for resources with existing organisms - leaving many affected humans unemployed and dependent on alms from the government.

It is true that population growth would probably be reduced in this kind of scenario - since usually the government pays the unemployed just enough to exist on.