Your doctor is totally wrong. He doesn't understand the FT3, does he. That is right scraping along on the bottom of the range, and it is that that is causing your symptoms. Whatever you do, do not accept treatment for hyperthyroidism. That will make you worse. You are hypothyroid.
I think your doctor is just looking at the FT4, which is high. ish. But the reason for that is that you cannot convert it very well at all. T4 is the storage hormone, and has to be converted into T3, the active hormone, which is needed by every single cell in your body.
But, even so, if you were hyper - and I don't believe in 'borderline' hyper, that's just silly - your FT4 would be much higher, your TSH would be much lower, and your FT3 would be way over range.
Your TSH is actually just nice. But with that low FT3, it should be a lot higher. It could be that you have a pituitary problem where the pituitary is unable to secrete enough TSH. But, you could also have the DIO2 gene defect which means you cannot convert. Or, you could have both. Either way, the problem is not with your thyroid gland itself.
I do hope I'm not confusing you, here. It is complicated for someone just starting out, I know. But please feel free to ask questions if there's something you don't understand.
So, what we have to think about now is your next step. There are several possibilities. a) ask your doctor to refer you to an endo to check for secondary hypo, where the problem is with the pituitary or the hypothalamus, not the thyroid gland.
b) get a DIO2 gene test - but, you can't get that on the NHS. If you have that defect, it means you cannot convert T4 to T3 (so are therefore dependant on the little that your gland is capable of making). But, you won't get that on the NHS. It would have to be private.
c) get your vit D, vit B12, folate and ferritin tested. It could possibly be that your conversion problem is due to low nutrients. This is rather a long-shot, but worth doing, anyway. Your doctor should be able to test those, if he's willing. If not, it's always possible to get them done privately. Details of private testing on TUK home page.
Sorry I can't be of more help, and sorry this is so complicate, but the most important thing for you to know is that you are hypo, and your symptoms are due to low T3. You are not hyper.