Hi Michael,
In all states except NY, NJ and RI, you can get direct access to lab testing. That is, you can order certain tests on your own without a doctor's order if you go to a lab that does direct access testing. At this link you can order and pay for your labs and they'll send you to the closest lab. Your lab results are available to you and you can submit them to your doctor. In some cases you can submit the testing fees to your insurance company although not all companies reimburse for direct access testing. It would be better to have tests for FT4 and FT3 since these levels measure your available thyroid hormone.
As long as both levels are within range, there isn't a specific balance between the levels that's needed. In fact, most people feel best with FT4 between mid- and high-range, and an FT3 around mid-range. Your thyroid hormone levels (in the absence of a thyroid disorder) are usually fairly stable and they're kept stable by the pituitary hormone TSH. So you'll see more fluctuations in TSH.
If you have a thyroid condition then your levels can differ depending on whether you're having a flare. In early disease, periods of flares and symptoms can alternate with periods of normalcy.
As for convincing your doctor, you could mention that the thyroid influences cardiac function, but he probably knows that. If he's tested you for TSH and it's normal, he probably figures that this is adequate, and in most people it is since normally TSH reflects thyroid function. As a screening test, TSH is usually adequate but when one has a thyroid disorder they're being treated for, TSH can be misleading. Best, Elaine