Child coming AND wife taking leave from work – what is the ‘trigger’ for us to move insurances?

Looking for help, not finding good answers from people around me!

I am essentially looking for if there is a legal reason or trigger that we would need to switch insurances from her work’s insurance (which we are on now as a family plan with us and our one other child) to my work’s insurance for all 3 of us and our new child coming

My wife and I both work

Second child is due Sept 13

My wife will be on FML after this date for 12 weeks and then officially on a leave from work for the remainder of the year starting around early December, and won’t be returning to work until next school year

We were told that once she is on fully unpaid approved leave, after FML, we will need to pay the school the difference of her portion of insurance that they are now contributing, which is around $900/paycheck (obviously not sustainable)

My insurance plans through work are much more costly and not nearly as good as hers with coverage benefits

My question is, do we need to switch to my insurance right when the baby comes or can we wait until she is on her leave (which would be most cost effective and beneficial to us to remain on there)? She is not being terminated from her job or anything and technically she is not ‘losing’ her insurance when she goes on leave, it’s just that it’s incredibly cost-prohibitive for us to pay that school portions cost. Ideally we would remain on her insurance until December then switch to mine. However, I just don’t know what is deemed as a triggering event for us switching if we can move at that time and don’t want to be caught in a situation where I missed the boat or am doing something ‘illegal’.