Based on your familiarity with those two labs, I assume you're aware of this site, but there's a phone number to call if you want a test not on their list.
(I used their service for my 12 week EOT)
(888) 732-2348
requestatest.com/
However, if you're willing to pay for a test out of pocket, have you tried simply going to a walk-in clinic and asking to have the test done?
Understand, for an insurance company to pay for the test they'll likely require a specialist's signature ... but if you're paying yourself, the lab shouldn't care who ordered the test.