Being hesitant to do overnights is different than denying the dad from seeing his son. I had to learn that one. He has had plenty of opportunities I assume, to see his child, so saying "no" to certain circumstances is not denying him. Does that make sense?
My ex tried to guilt me into believeing I denied him, when it was him, just like your ex, who was the absent disappearing dad, sceduling the visits on his time when it was convenient, etc.
You are the custodian I assume so you ned to set the ground rules on what is OK for your son. You do have that power. I would think that until he is a consistent dad and presence, then you could start considering doing overnights. Overnight entail routines, and he hasn't established any with your son yet.
I know it's hard. They've had the chance to get involved...like as soon as they know they were going to be dad. Why is taking this long? And are they really coming around? I think maybe it's best-- and I am talking to myself as well as to you--to let them practice a little consistency and consdieration for awhile bfore they get the privelege of being the caretaker dad.
I get upset about this, and I'm sorry to hear you are going thru this too. It's just plain hard!
(Oh my gosh, your son is only 6 months old!!! I would put my foot down. It's so primal, I know, but the baby needs to be with the mom for awhile. I breastfed til my son was 2 and at 17 months the dad guilted me into letting our son stay with him for 2 days. He told me to just pump, and didn't even care how it would impact our son and his expectations. I regret to this day still for letting him to do that to me and my son.)