From what I have read/listened-to, efforts by governments to incentive increased birthrates have little effect. This comes from data across different countries like Denmark/Norway & the USA.
Bronze Age collapse seems a little bit much ;D.
My bet is that the culture changes to adapt to the higher proportion of non-working age people. So more younger and older generations will move back in with each other in order to save on the biggest household expense (the house itself) and more welfare is directed towards caring for old people in tandem with the market responding by supplying more private care.
Europeans made it through ~1/3 of the population dying from The Plague, we'll make it through this _much_ less catastrophic demographic change.