Can you find a functional medicine practitioner (FMP) near you? This is basically what I am studying but not being an MD I have to be a nutritional therapist instead. There are very few FMPs here in the UK (Dr Rangan Chatterjee of Doctor in the House fame is one), but there are lots in the US where the movement started. An FMP will look at your whole diet and your medical history as well, of course, your thyroid condition. For example, if you have anti-thyroid antibodies, these are often caused by gluten because gluten mimicks your thyroid hormone structure to an extent and so the antibodies formed from exposure to gluten (which gets through your gut wall due to it causing release of zonulin which opens up the tight junctions between the cells that line your gut and this can allow gluten to pass through to the blood stream). Although avoiding gluten cannot completely reverse this situation it can help keep the anti-thyroid antibodies low and so may improve your ability to respond to thyroid hormone. There are several micronutrients that can help thyroid function too. An FMP would be able to help you work out a way of eating that suits you (and this might not mean going as low as 60g carbs though I suspect it would involve keeping carbs lowish and avoiding gluten).
The FM process is very empowering as you begin to see the big picture about how your medical history, genes, life events and diet all come together to cause the symptoms you are experiencing and this helps to see a way forward to reverse or at least reduce the issues.