R.J.
1) The less he sees / "has" his daughter, the MORE he pays in child support. You do NOT pay to see your kids. You pay for their care. Either when they are with you, or with someone else.
2) Very few parents are happy to see their ex during 'their' time with the kids. If your fiancé is just showing up whenever he happens to feel like it, MOST moms would be ticked off. It's disruptive, rude, self centered, and a lot more besides.
3) Even fewer would be happy to see their ex with their girlfriend in tow. Doesn't matter how nice you are. It's a pain in the neck for most kids of divorce to have to deal with the love interest of the moment... Which makes it a pain in the neck x10 for mom. Especially when dad has no interest in kids EXCEPT when his girlfriend does, or has already brought 3, 4, 20 women around over x years. Or during the few times he has the kids is shoving them off on said girlfriend, or is ignoring the kids for said girlfriend. Even if your fiancé does none of these things... This is what is going through a mom's mind. Totally legitimate until proved otherwise (doesn't sound like your fiancé has proved otherwise, yet).
4) To go from giving up his rights to wanting custody puts mom through the WRINGER. On both scores.
To me it sounds like dad either needs to step up and get something kind and consistant going... Or at the very least needs to stop putting this family through hell (pushing them away, showing up at random times, showing up at random times with girlfriend, pushing them away then demanding time).
Yes. Go for visitation. Not as a threat... But as the decent thing to do. If mom has full custody he needs to STOP just showing up, and get something consistent going that everyone can count on, and be there when he says he's going to be. Or to sign away his rights and leave them be. Or to not sign away, and pay... But stay away and stop this selfish attitude of showing up and disrupting their lives.