Your T3 seems to be in-line with your T4 levels, so perhaps T4 alone might work. But sometimes this really is wrong for some and T4 alone can be a disaster that most docs do not understand. Armour should work well in almost all cases, however, it has had potency problems and those affected who have tried Nature ? Thyroid have done much better on that. A synthetic T4+T3 product should be good too.
Thyroid hormone management can be quite complex/difficult for some.
When dosing these things, normalization of TSH and body temps are expected and these can guide dosing. Labs are very useful, but interpretation of numbers via statistical norms does not address one's own biology.
If lab numbers are mid scale and symptoms persist, one can then suspect rT3 interference, blocking the action of one's fT3. Low ferritin can also interfere with the ability of the cells to use T3. If cortisol is low, one can expect to find elevated rT3. So there are three other things that can be checked with lab work. Now you can see how this gets difficult and a bit much for the simplistic approach of many doctors.
Fixing rT3 also has complexities. And if one has been struggling to loose weight because of low thyroid levels, extreme weight loss diet attempts can lead to elevated rT3 which makes things worse, not better. Note that when one has elevated rT3, treatment with T4 or T4+T3 can increase rT3, making things worse. The root cause of the elevated tT3 needs to be addressed and T3 only needs to be used in the interim.
If one assumes that you should be medicated, one can take a simple approach to this and just do a trial dose to see if that improves things. If so, then you are motivated to go down that road and do further labs to improve the management of the condition. Some do advocate doing the other labs at the outset, but some docs or care management systems might not be helpful.
Given your symptoms, I think that you should aggressively pursue treatment. This then changes the focus to concerns about doctors and thyroid med options.
If this is confusing, it should be.