A lot depends on your insurance plan. If you have an HMO, you are probably out of luck. If you have a plan that allows you to choose your own doctor, you might have some luck. Even if the doctor doesn't submit directly to your insurance, some insurance plans will let you submit the bill yourself once you've paid, and
maybe get reimbursed. It's worth a try, at least, although it's generally a major PITA.
Which labs are covered often depends on your individual insurance plan. It all comes down to whether your insurance company deems the labs legitimate and necessary. It helps if your doctor puts the right diagnosis code on the lab request. If at all possible, you do NOT want the diagnosis code to be for CFS because the insurance company can legitimately claim no test is legit for CFS.
You want codes for gut diagnoses for gut tests, cardiac diagnosis codes for cardiac tests, and so on. I imagine Dr Kaufman is very good about coding labs properly to get the best possible chance for insurance coverage, but I don't know that for certain.