He does not need to help pay for you to visit, nor should you expect him too.
In some cases courts have parents split the cost of the children going for visitation. That would mean
BM and BF meeting half way to pick up and drop off the kids, or CP paying to fly the children to NCP home for visitation ad NCP paying to get the kids back home. (Or spliting the cost of round trip)
You say you miss your daughter, you say your ex won't let you see her, then you said that he offered to meet you for dinner or at the park. If you want a relationship with her and the possibility of getting visitation changed, you need to make the effort to go to where she lives and visit in any way you can. You should be responsible to pay for it as well. How can you tell a judge, I really miss her and I want to see her but only if he pays for it? A judge is going to want proof that you have changed and that if he goes to the trouble of changing the current order, you will have the ability and desire to act on that change.
You do not seem ready to do that. Remember, your daughter hasn't seen you in two years. I don't blame anyone who refuses to allow a child to go out of state to see NCP after two years. You need to visit her and get to know her again first.
I understand not wanting SM there when you visit. Maybe you could ask BF to meet you at a park and sit where he can see you and your daughter but not get involved in the visit. That is what we did the first time we saw my husbands kids after two years. We all met in a park. BM sat in her car and read a book while we enjoyed time with the kids. It was a nice start and after 5 more visits where
DH traveled to see them, and 8 months later he was granted his change in visitation request and the children have been coming to see us since. DH pays 100% of the travel cost involved.
You have to do your part and not expect others to do it for you.
KB