How does getting BAH work for an out of wedlock child who will be born in December?

In order to receive BAH, you must have a dependant enrolled in DEERS.
Since you will not be married, you will have to provide court order or paternity test proving you are responsible for this child. See DEERS required documentation at https://www.tricare.mil/Plans/Eligibility/DEERS/RequiredDocuments

As long as your child is in deers, you will get BAB with dependents.

I was in a similar situation once.

Basically............... look up your local DEERS office requirements to get a new dependent in the system. I presume the DEERS requirements will be the same everywhere you go, but the state the kid was born in is going to be different regarding how they give you the documents that DEERS is going to ask of you (my example is one kid born in Virginia and the other born in California).

Now, from there........ don't discuss any complicating family life situations. Just get the kids into DEERS. Give them what they ask for on the check list and move on in life.

At one point I did not claim them. We were unmarried, and the nuance to BAH is I would have to have a court order saying I agree to all that responsibility. The problem with that from my partner's side was such a court order basically predetermined I get the kids if we separate. So yea, as a drilling reservist where BAH is irrelevant she had no interest in that set up.

Then however, she messed up, and failed to renew her employer health insurance for the kids. So the kids no longer had health insurance.

Thankfully, because they were in DEERS I put them on my TRICARE Reserve Select rather painlessly, but I still wasn't getting BAH.

Life, having one complicates it.