Your FT4 is not all that low, it's true. But ... your FT3 is only 28% of its range. T3 is the 'active' hormone, and your level is just about half as high as the minimum level the majority of the population needs in order to feel well. For whatever reason, your body appears to be not converting enough T4 into T3 the way it's supposed to. That low FT3 and high-normal TSH are a huge red flags for the hypoT argument.
If you would be found to have thyroid antibodies, the deal would be sealed ... a diagnosis of Hashimoto's thyroiditis a certainty. Conversely, a negative result is not a sure sign that you don't have Hashi's; between 10-20% of Hashi's patients do not test positive for antibodies.
Finding an MD to treat does not in any way mean he would be a "crappy" doctor. And forgive me for saying it ... your MD is arrogant to insinuate that only an MD of questionable repute would treat you. The first MD I saw for my symptoms refused to treat me with a TSH of nearly 5, rock-bottom FT4, and multiple symptoms. The second one I saw confirmed my self-diagnosis (by testing antibodies, BTW) and began a treatment that ultimately didn't work out. The third one I saw prescribed the brand of med I successfully take now. Three doctors... one, like yours, said I didn't need treatment, two others said I did. Are they "crappy" doctors just because they didn't agree with the first one?
Here's a little thyroid history lesson for you. Before the late 1950's, before the TSH test and antidepressants were invented, people went to their doctors complaining of the very symptoms you have. Their bodies were examined for the tell-tale signs of hypothyroidism... not only symptoms, but signs. [Signs are different from symptoms in that they are observable, and so carry more weight ... for example, signs are -- slow pulse and reflexes, swollen ankles, puffy eyelids, scaly skin. Symptoms - mood swings, anxiety, brain fog, etc. - are reported by the patient herself and are not directly observable.] These people were given thyroid hormone without having to jump through hoops the way we do now. And guess what? They got better! If they didn't, the thyroid hormone was discontinued with no harm nor foul. In many ways, the TSH test has been a detriment to thyroid patients worldwide, because it has taken the art of clinical practice out of medicine.
The only way you could be harmed by a trial of hormone would be if it were given in too high a dose too soon, or if the dose would be increased too fast. Even then, harm happens mainly to the elderly. People with young, otherwise healthy hearts rarely have any problems. If it turned out you don't need the med, you'd simply stop it.
If by the longest shot imaginable you can convince this MD to give you a trial of hormone (doubtful), I strongly warn you that his statement about 'feeling better at first' and then your body 'returning to its current state' is false, and may be designed to intentionally set you up to fail. The true fact is that beginning thyroid treatment is commonly difficult in the first 6 months. Patients frequently - if not usually - feel better within the first 3 weeks, then regress during the next few weeks until the blood levels are checked again. This absolutely does not mean the treatment isn't needed at all; it means the dose needs to be higher! MDs like yours use this scenario to say, "See, I told you it wouldn't help. Here honey, take this Prozac instead."
I'm putting away my soapbox now.
Best of luck on your decision!