To my knowledge there's no historic precedent for what's happening. Times of regrowth in the past after catastrophic decline (such as major war) was generally already in the context of extremely high fertility rates. What we do have is increasing evidence, across multiple nations such as Iran and China, that it's relatively easy to lower fertility rates and extremely difficult to bring them back up.

So I don't think there's any reason to expect a bullwhip type effect. It could simply be that we've created a system which does remarkably well up front, and then catastrophically collapses in the tail-end. If one wants to be especially fatalistic, we could even be looking at the Great Filter of the Fermi Paradox, but I think that's improbable - there still remain many high fertility cultures and they will simply inherit the world as low fertility cultures die off.

Not only that, but there would be high fertility subcultures inside the general population that would be highly overrepresented.

It’s only possible to happen now due to birth control, so it is the first time. In the past after great deaths the mostly young population had a bunch of kids.

That’s impossible now, because the people left behind after the collapse are mostly old.

Birth control is probably a great filter.

In John Calhoun's mouse utopia universe 25 experiments, overpopulation inevitably leads to implosion characterized by asocial, juvenile behavior and inability to bond, and there was never any rebound.