Which insurance would be billed?

Our situation:

My spouse ended employment at previous employer on 7/10/23. My spouse elected to stay on cobra as she was pregnant and due to deliver in July. We paid the July Cobra premium which provides us with COBRA coverage (her previous employers insurance) through July 31st.

Our child was born on 7/29/23 and we were discharged from the hospital on 7/30/23. Since the birth of a child is a special life qualifying event, we want to get my spouse on my employers insurance now so we don’t have to continue paying expensive COBRA premiums.

My question is… which insurance would be billed for the hospital stay on 7/29-7/30 since technically the qualifying life event was 7/29/23 and that’s usually when the insurance with new employer would be eligible. However, she is also covered under COBRA at that time because we paid for the month of July.

Would it be billed to the COBRA insurance since that was the original insurance going into the delivery? Would it be billed to the new insurance? Ideally, we would like it billed to the COBRA insurance because we are already almost at the out of pocket max and we know the hospital is a preferred provider on her old insurance. We would have to pay a lot more out of pocket if they try and bill it to the new insurance.

I appreciate any help navigating this situation.