If you can not get GP to do these tests, then like many of us, you can get them done privately
Blue Horizon - Thyroid plus eleven tests all these.
This is an easy to do fingerprick test you do at home, post back and they email results to you couple of days later. Usual advice on this test, (home one or on NHS) is to do early in morning, ideally before 9am. No food or drink beforehand (other than water) don't take Levo in 24 hours before (take straight after). This way your tests are always consistent, and it will show highest TSH, and as this is mainly all the medics decide dose on, best idea is to keep result as high as possible.
If you have Hashimoto's then you may find adopting 100% gluten free diet can help reduce symptoms, and lower antibodies too. Selenium supplements can help too.
You need good vitamins for any thyroid hormones to work