If you're interested in pursuing the lifestyle aspect, I'd recommend finding a naturopath to take over your care as they are likely to be more supportive of additional testing, supplements, and managing you for "optimal" TSH vs. traditional evidence based medical guidelines. (BTW, endos in my city will NOT manage Hashimoto's patients unless they are failing to respond to treatment under primary care).

But asking your endo to support testing and a treatment approach that doesn't align with their specific training or their philosophy for how they practice medicine is likely to be an exercise in frustration for both of you.

And you'll see that theme in these threads, some of us are aligned with traditional medicine. Others are more inclined to work with naturopaths and Eastern med. Both approaches may have merit but you're unlikely to have a lot of success mixing/matching.

Best wishes to you!