If you Op have given him access then that is proof that you believe he is the df. Putting him on the BC or not doesn't make a difference if you allow contact because he can just apply to be added on.
You need to stop communicating with him when he's acting arsey or rude. Stop engaging in it, and stick to what you want. The lo is small and contact should be safe for you and the dc.
If he wants and dna let him arrange and pay for it, if he wants court again let him arrange and paid for it. Cause he either will or he won't.
Stop putting yourself through this stress because at this moment the lo only cares about you.
Of course it's great for a df to have a bond but this lo isn't a tool to fight over.
Tell him he can visit on x day at x time and that's it, obviously don't pick times he working, and twice a week for 30 minutes to an hour is better. Little an often but you of course can be there to supervise - watch some tv while he does his thing, DONT play on your phone that will cause tension.
You want to be helpful and supportive but you don't have to sacrifice.
Only you know if he has the money or energy to actually go through mediation, dna and court.
OP I just had court with my ex today, hasn't see his ds in 6 months, it was for a non mol. And he moaned about not seeing his ds but when asked what had he done to see him - it was a big fat nothing. He harasses me but can't actually use that energy to see his ds.
It's all BS. Manipulation. My ex actually started crying on the phone as if that would make the judge feel sorry for him 🤦🏽♀️ it didn't I got my order.
Remember it's about you and your little one.