Obviously I don't know you so can only comment on the picture you've described. You sound like you're trying to play "happy families" in a situation that is impossible to fulfil that dream for you.
It strikes me that you'd like your "perfect world" to involve everyone responding to your "encouragement" for everyone to come together in perfect harmony, and while I am all for people being as peaceable with each other as they can be, I can't help but think of the true saying: you can take a horse to water but you can't make it drink!
If a man has not had contact with his kids for 2 years, and has made no effort to change that, he really clearly doesn't want to bother.
It would be a mistake for you to try and change him. You will only wear yourself out trying and it will never work. You say "he doesn't have the balls to fight for them", but I'd say realistically its more that he doesn't have the interest. You fight for children if they're important enough - balls or not.
It would be really hard for those kids to have their Dad reintroduced to their lives, only to lose them again. Or worse, kids have an uncanny nose for smelling disinterest and can you imagine being stuck with a visit from your Dad knowing full well his heart isn't in it?
You mentioned the kids would have loved to meet your new daughter. Showing off your new baby is not a good reason to mess about with the emotions of those poor kids. Besides, two years is a long time and you may find the interest they showed back then, has been lost by now.
It may be hard for you to face this, but it does seem to be the truth of the matter. I'm sorry life isn't as rosy as we'd all you'd like it to be.