Let him take our son or not?

Long story short I had a baby boy in March. My sons father left me at 10 weeks pregnant and moved with another women. Didn’t support or help me my entire pregnancy. Didn’t help prepare for our son. I had a scheduled induction he was invited to and he decided to leave and go on vacation with her. I allowed him to come meet the baby briefly at my house once he came back. He came and called CPS on me (false claims & to be petty) and also sneak did a at home DNA test thinking these things would give him his rights because he never showed up to sign the birth certificate. After seeing he failed he disappeared for three months. He recently contacted me and wants to be involved with our son. I said he can visit in my home but he refuses he won’t see him unless I bring him outside and he gets to take him for the day or night.

Should I allow my now 7 month old to spend be day alone with him or have a sleep over???

I don’t know I don’t feel comfortable but at the same time I don’t want to take away my son having a father.

Any mamas please give honest advice.