I think this relationship holds a lot of significance for you, as it symbolizes some of the things you disliked about your father and still carry around with you. But from their perspective, that meaning and history isn't there. I think they probably just see you as a cousin who they don't really know much about.

If you approach this just from your perspective, apologizing for your father or getting your own personal history involved in another way, I think it might weird them out. I'm not saying you could NEVER do this, but I think you probably want to establish a little rapport first.

I get that it seems like a puzzle right now and you're trying to figure out which angle to build that rapport and reach out. I think it looks complicated because of your own history with your father and because of some personal issues with rejection that you alluded to. I think it's simpler than that, though.

I'd suggest pretending like that wasn't an issue and just getting to know them as if they were estranged family members you just didn't really know. I'm confident that you would know exactly what to say to someone in that situation, something simple like "it's great to get in touch with you finally! What are you guys all up to these days?" If a relationship forms, it'll form naturally over time. It's not something you have to rush or even work at.

Good luck.