Make sure that you have any future blood draws as early as possible - I aim for 08:00. For that is when TSH is usually highest.
It would be good to have the reference ranges, especially for FT4, as that varies widely from lab to lab.
I agree an antibody test is appropriate.
The inter-relation of stress and thyroid is itself complex and poorly understood. That there is a relationship seems clear - but not that reducing stress would, in and of itself, reduce TSH and increase FT4.
You do have a problem with an unwilling GP. Any chance seeing a different GP? A locum when she is on holiday or another doctor in the practice?
Continuing to have elevated TSH and low-ish FT4 is not good. My take is that it can be sensible to do repeat tests to ensure that things do not resolve after a virus or other transient issues, but when it is like that repeatedly, there is something which needs treatment.
As always, also try to get the following checked:
(Though you might have an uphill struggle to persuade your GP.)