Denial of newborn care question

I'm trying to figure out how to navigate this situation and would appreciate any advice. My wife and I (father) both have healthcare through our employers- both self-only until we had our child.

Our baby was born in January in the hospital via C-section. For various reasons we had decided we'd be adding baby to my health insurance (change to self plus one) rather than my wife's. I did so immediately upon birth of the child and baby has been on my plan since that time- all routine visits, sick visits, etc have been covered.

I just received notification that my insurance will not cover the cost of baby's 4 night stay in the hospital (~12k USD). They cited that coverage is provided for newborn routine care when the mother is a covered member, but since the mother is not covered under my plan, my plan will not pay for the baby's newborn hospital stay.

My wife's insurance covered her 4 night stay (also ~12k USD) plus all the surgical costs for the C section and other parts of labor/birth etc.

What should we be doing here? Is there recourse w/ my wife's plan to get them to cover this? Should we be contacting the hospital at this point about where they are sending this bill? Is there any tactic or line of enquiry I should be pursuing w/ my insurance company?

This is so frustrating as if we'd known this we would have just put the baby on my wife's insurance for the first year then switched her later. But we had no idea this would be an issue.