I think it's likely that about the only testing you'll get at a free clinic is a TSH test. Unless that result is completely out of range, it's unlikely they would provide any treatment. They wouldn't be the first place I'd think would pick up on all the nuances of thyroid disease. But as McA says, it certainly won't hurt to ask.
I just read your conversation with accessn12 (Helen) regarding your previous labs. She said she thinks you probably aren't hyper but have a thyroid binding protein quirk that's probably harmless. I tend to agree that you probably aren't hyperT.
There are online labs where you can self order blood tests for thyroid. A very popular and reliable lab that performs self ordered blood tests charges about $85 the Thyroid Panel II which includes TSH, free T4, and free T3. If you only want to verify that you're within range, this is an affordable way to do it. Several states ban the use of these labs though - NY, NJ, and RI - there may be others. So that may not be an option for you.
I think I'd have a frank conversation with your general practice doc. Just tell him you don't think you can swing the cost of an initial visit to an endo plus tests. Unless you're having significant symptoms - based on the tests you already reported here, he might agree that it will be ok to just retest periodically to keep an eye on things.