As PPs have said, this is about what's in your DD's best interests.
Firstly, it's in her best interests for him to pay you child maintenance, because even if you don't need it, you could put it into a savings account in her name. So accept his offer to pay maintenance. If it's a conditional offer only (ie he thinks his child is "pay per view") you'll have to go through CMS, but bear in mind that it can be easy for self-employed people to avoid paying.
Secondly, is it in her best interests to have contact with a "father" like this?
"He has never demonstrated any trustworthiness, empathy or responsibility."
I don't think so, personally, but I can understand the desire to give him a chance.
Luckily he has no legal right to any contact atm, because he's not on the birth certificate so he doesn't have parental responsibility. He could apply to the court to get it but I don't know how likely he is to a. bother and b. succeed, based on showing zero interest until now.
You could meet up with him, see if he is willing to meet you a few times first - will he stick to it or will he lose interest when he doesn't get what he wants immediately? If he does stick to it and seems to be respectful towards you and DD's needs, perhaps the next stage would be for him to have communication (but not face to face contact) with her, eg he could write her a letter, you could see if she wants to write back (she doesn't have to!) and if he is consistent enough to write to her regularly, then maybe they could meet but it would obviously need to be supervised contact.
If I had to bet I'd say he won't get that far, he'll get frustrated and revert to his usual dickhead self.