If you're a single parent and your child's father is a formally abusive piece of work, lives in a different state than you and the child, has supervised visitation every couple of months, what happens to your child in the event that something happens to you? Does the child automatically go to the father, or to your family, or to a foster home while the father and your family endure a long and bitter custody batter, or what? What if your child has an adult sibling that they are also living with- does that adult sibling have a change of custody, or is it only the father who does, regardless of events of the past? Is there any way to make your fiance guardian, or at least in the running if, God forbid, anything happens to you? Is there any way to guarantee that the child does not go to his father? Anyone have any experience with this?
Spinoff of the choosing a guardian in the even of death thread...
^^ Yeah. :-(
Are you still working on your divorce agreement? If so, it might be something important enough to consider there. It wouldn't actually be enforceable (as in, if something were to happen to you, the Court certainly could decide it's in the children's best interest at that point in time to reside with their father, or any other person for that matter, even if you and he agreed that your sister/cousin/friend would assume custody of the children) but having it written down in an official place might still influence that agreed outcome because it gives him an "out" (IIRC you said he doesn't want to be a full-time parent anyway but maybe I'm mixing up mamas). Either way you should have a Will that identifies your preferred custodians in case he dies and then shortly after you die (if he dies first, your Will takes precedence).
You should continue to record evidence against him being a primary parent that your preferred custodian can access in the event something happens to you. If they're going to fight custody defaulting to him, they'll need unbiased evidence to demonstrate why it's so overwhelmingly not in the children's best interest to reside with him that they should ignore his right to parent.