That's assuming the economic arrangement during and post climate change is conductive to rising birth rates in the first place.

What people, especially on the right, don't realize is that falling birth rates is not a matter of incentive or culture. It's correlated with rising literacy rates. That's it. The moment people realize the costs of raising a child in an industrial society and how that affects their future livelihoods they simply don't. The reason birth rates were high in pre industrial societies is because having lots of children was the economic strategy of every living human - from peasants to landlords to kings and merchants. The reason we had exploding growth rates during the industrializing period is because literacy rates and standards of living actually went down in many places, and didn't catch up to the industrial reality until much later. Now, having a child is a luxury for the gentry few. Therefore, populations de-grow.