Hun, I don't really want to comment much on your ex showing off about being a Daddy when he's taken no interest, as wouldn't know what to say apart from what the other girls have said - His behaviour would make my blood boil tbh! I'd be thinking about applying for comlpete custody in terms of not allowing him access at all - If the only reason he wants it is to be able to show off, he doesn't deserve it in my opinion! Your baby will be better off without a Dad if that's the only interest he has in her!
What I did want to say, is that you mention financial help being a 'no go' - However, he has a legal obligation of care - at least financially - to his daughter, so be sure to insist he pays up! If he refuses, inform him that he will be breaking the law if he doesn't contribute, and that it would be better for him to settle it outside of court, because if you go through courts he may even end up paying you more! This link might be of help - it outlines how to calculate and obtain financial child support fomr the non-costodial parent to yourself: https://en.wikipedia.org/wiki/Child_support#Obtaining_child_support
Best of luck, hun