I found the easiest way to do it is just to ask for the specific tests you want, come armed with information coming out your ears, and use it in conversation. Show him your knowledge. If he is the kind of doctor to get angry about that, he deserves to be fired.
If all else fails you can get the tests you need independently.
http://www.healthcheckusa.com/
VIP Profile Plus II does TSH, Free T3, and Free T4. I think they also have a more detailed thyroid panel that does antibodies and such - look around and contact them to ask if necessary. It's spendier but sometimes you can get your insurance to reimburse you.