Don't worry, you haven't made up anything about Hashimoto's.
TSH and actual thyroid hormone levels do not have to be out of range in order to have Hashimoto's. The doctor who told you that is wrong and does not understand how it works. Many of us have it for years, with the antibodies destroying our thyroid gland for many years before your actual thyroid function levels go out of range.
Do you have the actual report from which you posted the results above? It would be most helpful if you'd post the reference ranges that match the Free T4 and Free T3 since the ranges vary from lab to lab and have to come from your own report. That will help us compare your results to the ranges and see exactly where your levels fall within them.
I'd recommend that you go back to the doctor that wanted to put you on the NP; that seems like the one most likely to help you feel better. Just make sure s/he's willing to increase the dosage adequately, as needed to alleviate symptoms.
I would concur that the TPO antibodies that high would suggest Hashi's.
Going gluten free may or may not have an effect. But only if you are gluten intolerant or sensitive. Any gains of feeling better will be almost entirely because of the gluten intolerance. it will not reasonably affect your thyroid condition. If it does, it would be extremely minimal in my opinion.
Some believe that going gluten free helps minimize the antibody response. That may or may not be true. from what I have read, there has been nothing conclusive of this, and may be of limited help for a period of time.
What symptoms do you have? How uncomfortable are you with low thyroid symptoms? If you are not feeling poorly, I can somewhat understand a wait and see attitude. But if you are symptomatic, then that is different. At least in my opinion.
What are the results of actual thyroid hormone tests, along with reference ranges? It's entirely possible to have symptoms of hypothyroidism, even if hormone tests show levels to be "in range" if levels are in the bottom half of the ranges.
Please post those results, with ranges and we can go from there.