You can tell her that you know about FT34 being the most biologically active thyroid hormone, and that it largely regulates metabolism and many other body functions. You can also mention that scientific studies have shown that FT3 correlated best with hypo symptoms, while FT4 and TSH did not correlate very well at all. You can say that you are aware that if FT4 is known, FT3 can be estimated fairly well, unless your body is not converting T4 to T3 very well. In view of its importance, why not just test for FT3 and be sure. You can also say that the accuracy of current test procedure for FT3 is more than adequate.
You might also find this link to be of value. It is written by a good thyroid doctor. the letter is sent to the PCP of patients consulting with the author of the letter, to help with their treatment.
http://hormonerestoration.com/files/ThyroidPMD.pdf
If your doctor still refuses to run FT3, then remind her that you are the customer and insist on FT3 testing anyway. If all that doesn't work, then you probably need a new doctor.
Let her know that you would like it done to see how you are converting your free t4 to free t3, as it is very common for women to have conversion issues, which in turn can cause residual hypo symptoms even though your tsh and free t4 say you are optimal.
Free T3 is the most active form of thyroid hormone in the body.