Unless you're a lawyer, you should be saying "they" as opposed to "we".
The complications would relate to proof -- they need to focus on getting their evidence together. The midwife may have records that can establish the date and circumstances of the birth. Depending on the full context, availability of other evidence and the expectations of the court, it may be appropriate to get DNA testing through a court-approved facility to prove the biological relationship.
The divorce court is not going to ask to examine the child's birth certificate.

