I would absolutely pay for it. ALJ's aren't stupid, they're well aware some doc's will make claims that they can't substantiate and having this type of testing will give you evidence that neither you or anyone on your behalf is exaggerating your limitations. This is especially the case with an invisible disability.

Can you make your claim without it? Perhaps, but IMO it's not worth the gamble, unless the cost is such a burden you'll be having a problem covering your necessities.