You are correct that in most cases a Free T4 is best for monitoring treatment. Sometimes in conjunction with TSH. It may be that FT4 is not immediately available at the lab where your testing is performed, which may influence her decision to order it. I would have a frank discussion with her about your concerns and let her know you've been doing some self education on the subject. A good physician will not feel threatened and actually respect your desire to be a proactive participant in your care. As far a getting copies of your labs, you should only have to sign a release to get them either personally or sent to another physician.
Good luck!