This seems to be upsetting you. Have you talked to your husband about it? If you need a definitive answer I suggest you speak to him. The important thing here for you is to communicate with your partner, explain your fears and feelings and hopes and suggestions. Then there can be some open dialogue between you about what is the best way forward.
I suggest proposing the idea of setting a clear agreement in writing (email would do) to his ex, stating that unless she takes a paternity test that he will not provide any more money. If she refuses then it is clear. If she takes the test hoping she's wrong about the dad, and it turns out he's not, then that's helpful, his requirement to be involved is removed. However if she takes the test and he is the father, then to stop arguments, I would suggest a set monthly amount by standing order, with instructions about set times in the week/month (if he wants this) that he can see the child and spend time with child.
If you find you need arbitration, there are centres in the UK that help with this, I'm not sure what is the process in the States.
All the best,