Dual Insurance for infant (exclusive plans for each parent)

My husband and I work for competing healthcare systems with very exclusive strict coverage. Ex: I would not be able to see providers or receive care at a hospital of his institution without outright denial as it is considered out of network and vice versa.

I gave birth to my son in my hospital and my husband added him to his employers plan soon after his birth. We were under the impression that my son would be covered under my plan for the first 30 days after his birth (which I found out later is not true.) Because my hospital would be considered out of network with my husbands plan we added my son to my plan with the advisement of each insurance carrier.

However I just found out about the primary and secondary insurance rules specifically the birthday rule. This would make my husbands plan primary for my son. How would this affect coverage for the hospital bill? Will my insurance cover even as secondary?

They are still processing the hospital bill for my son. Will I be able to drop my husbands plan anytime and keep baby on my plan?