If you haven't already, please get yourself over to the thyroid section of the Health Unlocked forum. Should be able to find fairly easily, I think there is also a link on the Thyroid Uk website (another extremely helpful website).
There are some very knowledgeable people over there. As has been mentioned on here, GPs know very little about thyroid disease, and even less about Hashimotos. Even endocrinologists, who should be the experts, don't seem to know that much about it, and the majority of them are diabetes experts rather than thyroid experts. Unfortunately, you'll find that you need to become an expert in your own condition. The Health Unlocked forum can definitely help with that.
My TSH was similar to yours at the start of my journey. I had to do a medichecks blood test to get all my thyroid labs done, including antibodies, which showed than my underactive thyroid was down to Hashimoto's disease. I had to bed to be started on levothyroxine, and was started on a stupidly low starter dose of 25mcg. This was increased every 8 weeks, as I felt no better (in actual fact, I was pretty much bedbound for a year with how awful I felt). At 150mcg, my TSH went under the bottom of the range, but my T4 and T3 were still low, showing that I needed an increase. However, GP said her hands were tied as my TSH was out of range, and that's all they look at.
I had a lot of help from Health Unlocked, and started on T3 medication alongside my levo. This helped a lot. I'm a few years down the line now, and in the last 12 months have moved to T3 only medication, as levo really did nothing for me. I self sourced for a long time (you can buy it over the counter in pharmacies in Greece for 2 euros a box, I also ordered online). I was referred to an endocrinologist when I had already started T3. He wasn't happy, but he did also listen to me and my symptoms, and how they'd improved with T3, and we came to a bit of an understanding.
I did eventually manage to get him to prescribe it to me, then had another fight to get it prescribed by my GP after 3 months (they ended up doing an individual funding request). It's been fight after fight, and really hard work. I'm lucky that my husband is quite science minded, as has been right there beside me helping me fight.
I'm now on 75mcg of T3 per day, which I take in one dose. I believe I have thyroid hormone resistance issues, which is why I need such a large dose. I'm still not completely well, but I can function, and compared to when I was bedbound taking levo I'm a million times better.
The GPs don't understand how to read my blood tests at all, and the last two I've had they've called me in a panic. My TSH is barely perceptible (to be expected, especially being on T3 medication), and my T4 results were pretty much zero (again, to be expected, as I'm taking T3 medication rather than levo which is T4 medication). When I explain it to them they're fine, but shows they really haven't got a clue. I also have to fight to get my T3 tested, as nowhere likes doing it, but given I'm on T3 medication only it's probably the most important test for them to do!
I also went gluten free 3 years ago. I did have the coeliac blood test at the time, but didn't realise I wasn't eating enough gluten (should be the equivalent of 2 slices of bread for 2 meals per day for 6 weeks prior to testing) and of course it came back negative. There is a lot of evidence out there to show it can help people with Hashimoto's disease though.
Sorry for such a long post. If anything doesn't make sense, I've just come out of hospital following emergency surgery, so I'm firing on even fewer cylinders than usual!