I would go with internal medicine before an orthopedist. I think you’d have a better chance of that doctor having knowledge closer to what you need, given the range of possibilities of what is causing this.
Have you tried an elbow brace that goes around your forearm? I thought it couldn’t possibly work, but it did. It doesn’t impede movement at all, unlike a brace on the joint itself.
Do you mean one of those straps for tennis elbow? I have one and use it when I’m doing “something,” like mucking stalls, yard work, etc. It helps, but it hasn’t alleviated the overall discomfort.
For short term hand pain relief you might try a pair of heated mittens when you’re sleeping or resting. There are hand therapy mittens that are microwaved and others that are electric. Both have given me relief but the microwave type only lasts a couple of hours and the corded electric version gets me wrapped up in cords if I try to turn over.
Have you looked at the ILADS physician directory? These are physicians who handle a lot of tick borne diseases.
Also, Lyme testing has some significant issues. There’s a window to test where antibodies are high enough for a positive result if you’ve been infected.
If you test too early the antibodies may not be high enough for a positive test. And if you test well after the initial infection they may have dropped. The CDC doesn’t recognize chronic Lyme but a lot of people have symptoms well after the initial infection.