Please don't take this the wrong way, but I think it's probably a good thing that your child has supervised visits with her father.
If you are going to be this critical when he's under supervision, I shudder to think how things would be for him if he wasn't.
The visits are short and supervised. He hadn't had contact with his daughter since she was 8 months old. I understand your anger and/or concern over this, but if he gets about an hour with her a week, and it's been 8 weeks, that adds up to about 8 hours he's been around the child.
Yes, he is her father, but you have to stop and think that it might be uncomforable for your daughter to have a man she is just getting to know checking her pants or taking them off to change her.
He brings supplies. It doesn't sound like he's unaware that she needs diapers, etc. But, if the center has a policy about fathers changing their daughter's diapers, it's the center you should be dealing with instead of assuming that dad doesn't have common sense.
Yes, diapering is a part of parenting, but you'd be surprised how many men who are in happy and stable marriages have never changed a diaper. I don't think it makes them unfit parents.
I mean no offense by anything I'm saying. I just think that you have some bitterness about him coming back into your daughter's life and you are expecting him to cram all parental responsibilities into one hour at a time to prove to you that he can be attentive.
To you, it seems like all he's doing is having fun when that shouldn't be the point.
However, supervised visits aren't fun. Not for the parent being supervised. Watched, judged, possibly criticized for any possible thing.
I DO get the diaper thing from your point of view to an extent.
But, I bet you are a mom who would never let a strange man see her daughter's privates.
Just because the man you are talking about IS your daughter's father, by your own admission, he's been a stranger. He's trying to correct that. An hour or so at a time. If the center doesn't want the liability of being accused of allowing any impropriety, then you will have to go to court and tell them that you are not happy with your daughter leaving supervised visits with a wet diaper.
Maybe the court will change who does the supervising.
Best wishes.