Should I allow him to be involved with baby?
Of course you should allow him to be involved with his own child, why would you not?
I agree with the PP, it sounds like he left you to get back with his ex.
It’s very hurtful but now you need to look towards the future.
Do not let him back, some men will flit between their kids mums and you don’t want to get into that.
Do you have someone mutual that can pass on messages?
I think for the next few weeks it’s important to not speak to him directly.
You should only see him whilst you’re giving birth/after the baby is born, depending on your preferences.
Then he can have contact say 3/4 times a week for short periods in your home, but try and get someone else to do the handover whilst you go and have a bath or something upstairs.
In the future, you can start arranging the contact yourself and start to co-parent properly but for now I’d try and avoid seeing it speaking to him.