Trog - it's very likely that your GP is considering only Primary Hypothyroidism where TSH is raised alongside low FT4 and FT3 and where the problem is with the thyroid.
For low TSH with low FT4 and FT3, look into Secondary Hypothyroidism where the problem is not with the thyroid but with the pituitary, or even Tertiary Hypothyroidism where the problem is with the hypothalamus - one article drsaulmarcus.com/thyroid/5h...
It's unlikely that your GP will know of these, maybe ask for a referral to a knowledgeable endocrinologist who specialises in the thyroid rather than diabetes.
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