Child support is an epidemic problem in the United States due to so many fathers running away from their responsibilities. However, you have a neither here, neither there, type situation in which the father seems to be willing to pay something, but not everything.

It sounds like you keep having to negotiate with him and reach a compromise, every time you come to a snag.

I hope you will agree that it's not possible to get blood out of a turnip (meaning that if the father doesn't have the money, how can you get any?)

It's sounding like he has to sell his entire house in order to pay for the braces. This is something the court will not make him do.

It's also sounding like your present attorney might be getting weary of going back into court again and again.

It's very easy to switch attorneys. All you have to do is request that your attorney sign off on the case. There's a one-page form (which he has access to) that he'll sign, taking himself off the case.

Then you could get another lawyer.

However, your NJ lawyer's advice about approaching the NY court where the father lives (and all of your lawyer's advice for that matter) is good advice.

Why not keep your NJ lawyer and then go over to NY and see what results the court will give you there? But before you do, call the NY court (or get legal advice on the subject from the NJ court) as to whether the NY court will take your case, or whether it will just send you back to NJ again.

Probably the NY court will take the case because the father lives there, giving the NY court jurisdiction over the father and making it convenient for the NY court to enforce its orders.

Braces can be a part of child support, or not.
First off, dental braces are not a medical need, they are an elective cosmetic procedure that over 90 percent of the population does not get.
But one could argue that this is needed as a function of overall health.
Talk to your attorney, but it sounds like you are pushing the limits already.
I work divorce law and can tell you that I would not have any trouble whatsoever in getting braces as a part of child support thrown out.
I will also tell you in all honesty that I make my money on new divorce cases, child custody and finding delinquent parents, I would not want to take a case where it takes a lot of time, preventing me from finding lucrative cases.
Please understand that an attorney is running a business.