The exact specialism doesn't always mean much by itself. My local "fatigue"/CFS clinic is based in and fronted by immunologists, but they're job - how they see it - is to do the arbitrary tests to exclude you (they likely see anyone with Oxford-esque "unexplained fatigue") from having a biomedical immune disease, and recommend a "training" programme of GET, which they were doing even before the first GET study (by Sharpe IIRC?). People becoming disabled and requesting support with social services or testing for non-diagnostic abnormalities is a horrific unreasonableness to them.
For years they had a web page stating that CFS has "no abnormalities", can be caused by a marriage breakup, is pretty much all about "stress" and deconditioning (though a few people have viral infection of muscle, but not enough to contraindicate GET). After an "asessment" if you report that GET or just trying to keep going (as naturally they don't recommend resting for severe cases as they automatically assume everyone's been spending weeks in bed from the start!) is not working or worse, they simply have no more interest and you're written off as an impossible psych patient. Most pwME take a dim view of them except the local support group the committe of which is stuffed full of people who think they can do a "deal" with anyone and everone.
I sometimes wish I'd gotten sick ten years earliert then I could have had a chance at getting IVIG, or amantadine which I'd not heard of before.
Now the situation is that you could be have late stage progressive ME but you'll have to buy your own best hope from an internet phramacy unless you have pockets deep enough to see a prvate doc.