Yeah, well, politics. In the U.S, it's usually diagnostically going to boil down to either Lyme (acute vs late stage), or Post Treatment Lyme Disease Syndrome. Practically speaking, it shouldn't matter to patients since ultimately all we need is a doctor who can diagnosis and treat - and treat outside of prescribed norms if deemed necessary. In many states, that is not going to be easy to do. Also, the reality is that it DOES matter, because once you're slapped with the PTLDS tag, it's devilishly difficult to find treatment, or more precisely, anybody who is willing to offer treatment options.
If I were keen on being treated at a specific facility, I would simply suggest that I have been refractory to treatment to date (if I'd in fact received an IDSA recommended regimen), and have late stage, and ask straight out if they would be willing to try further and different regimens. I fear you will discover many will not be willing to do so.
Or I'd pick a reputable physician - it doesn't have to be an LLMD, but odds are it would be - that is willing to color outside of the lines. There are several of these in striking distance of both Columbia U and SB. In Hudson Valley, NY, 50% of ticks tested recently came up positive for Bb. So there's a bunch of clinicians over there that have hung up a Lyme shingle and are well versed in treatment options. I would imagine what a doctor such as this would do is give you a physical, run blood work and decide if they want to run an MRI or SPECT, then send the draws to SB or IgeniX or a lab they trust, and send you to Columbia U or whatever to execute the MRI. I am told it DOES matter who does the MRI or SPECT, in that interpretation is everything. But I'd urge caution: not all of these doctors are necessarily copasetic.
Incidentally, even though it's a political no-no to call anything Lyme "chronic Lyme" in the U.S., the NIH has at least one long-running study that identifies the patient cohort as chronic Lyme patients. That's only because the test was launched way back in the 90's, predating the reluctance that currently envelops the term.