I've been divorced and have remarried. Ah, the kids. They are the reason that remarriages with children involved fail so often. Having said that, I am very happily married for 8 years. We have had andcontinue to have rough times where the kids are concerned. It is the nature of the beast in some respects, I think.
Some things to remember are: they are right, he is not their dad and they did not ask to be put in the situation they are in now but are being asked to deal with. It really doesn't matter what led to the divorce in the kids minds all they know is that they aren't a whole family anymore no matter how crappy that family may have been.
Having said that, it takes a very strong person to put up with the stuff that other people's children dish out. I cannot tell you the number of times I was hurt to the core by DH's children. I cannot tell you the number of times I questioned whether or not I should stay in the marriage. I cannot tell you the number of times I gave up on DH's kids. Lets just say it has been a bunch. Let me also not that DH had/has custody of his kids so they were/are always with us.
Having said that, one day I just let go of trying to control everything and let God do it. It was such a relief to get that burden off my back. Around the same time, my husband realized that I wasn't the enemy, that I actually had some good/great parenting skills and that I was there to support him in his decisions and would give him some advice if he wanted it. I also made sure that I complimented him whenever he did what I considered to be a good parenting thing. I also expressed appreciation to him in how he treated my son and the things he did for him even though he wasn't his father.
Now, I know you are not married that you are just dating but I said all of that to you to help you think about the fact that it is a very difficult position to be the kid and to be the significant other trying to work your way into their world. Obviously you, as the mother, should expect appropriate behavior from your children; but you should try to understand where they are coming from--I'm sure they feel threatened (will they lose you too?), they feel disloyal to their dad if they even think about maybe liking this other guy (even though they can't stand their dad) and who knows what else.
The man you have in your life has to decide what he is willing to put up with. How much he loves you, how much he understands just where the kids are coming from and how much he can grow to love them. He has to be real thick skinned and shouldn't try to force what isn't there but appreciate what is there. It is the moments that count I think in these kinds of things.
One other thing I did when I was divorced, is no man ever met my son unless I felt that this was a relationship that may go somewhere. He had enough loss in his life when his dad left so I tried to be very careful in that regards. I also tried to be very discreet in the public affection area so as not to make any of the kids involved uncomfortable.
Sorry if I rambled on and didn't address your question as you wanted, this is just the stuff that came to my mind as I thought about your topic.
I wish you the best in this situation. It is difficult for all of you and hearts are so tender. Take care.