I think the GP needs to go back to school. You need to have a Free T3 test (at the very minimum) before your GP can diagnose you as hyper, it can't be done just on the basis of a TSH test, which is clearly what he is trying to do.
With your free T4 being so low in the range I cannot imagine how your Free T3 could be over the range (which would indicate you having hyperthyroidism).
The thing that really needs to be investigated is whether you have secondary hypothyroidism i.e. perhaps your pituitary is incapable of producing TSH because it is diseased or damaged. Your GP is not the person to do those investigations, for that you need to see a good endo.