I agree with your assessment. Free T3 is used by every cell in your body. Also, Free T3 correlates best with hypo symptoms, while Free T4 and TSH do not correlate well at all. Your Free T4 is only at 12% of its range, when optimal seems to be mid-range, at minimum. Your Free T3 is only at 30% of its range. It needs to be in the upper third of the range, or as needed to relieve symptoms. So I would push to get both increased.
In addition, hypo patients are frequently too low in the range for Vitamin D, B12 and ferritin. Low levels can also cause symptoms that mimic hypothyroidism. You should also make sure to get those tested and then supplement as needed to optimize. D should be about 55, B12 in the upper end of its range, and ferritin should be about 70.
One last thing for now, is that I suggest that you should also ask to be tested for Reverse T3, when next tested for FT3 and FT4. Reverse T3 is a mirror image molecule of T3. It is a normal product of the conversion of T4 to T3; however, under some unusual conditions excess RT3 can be produced and that can block the effect of FT3. So it is good to get a baseline test for RT3, to determine if excess RT3 is a problem or not.
If the 3000 IU dose of D3 resulted in a D level of 73, then that dose should be reduced. I don't know if 800 IU is adequate. You need to re-test and find out.
B12 should be in the upper end of the range, so there was no need to stop supplementing. Your ferritin level may well have been high due to the intravenous iron. That effect should have diminished by now, so I suggest that you should test and find out what it is now and see if daily iron supplements are required.