Your T4 is rather low, regardless of reference range, but it would be handy if you have the reference range so we know just how low it is.
Going by your TSH, you are fine but we know that TSH can be misleading. Your T4 does need to be higher than that, but you will find it difficult to persuade your endo of that. You may have more luck with your GP. Dr Toft states in his book "Understanding Thyroid Disorders" that T4 should be towards the top of the range or even slightly high when on thyroxine and that TSH should be below 1 or even suppressed and that this is fine provided your T3 is within range.
Dr Toft is former chair of the British Thyroid Association, the body responsible for the current (appalling) thyroid treatment guidelines, and the book is published in association with the BMA so your GP shouldn't really argue with what he says. The book is only a few pounds so it may be worth getting it. I don't agree with everything he says, but that one snippet can be quite useful for getting the right dose of thyroxine.
Other things your GP needs to check (if your endo is convinced it's not thyroid) are:
Ferritin ( should be above 70 for women, higher for men)
vitamin B12 (should be above 400)
vitamin D (deficiency is very common in the UK)
I hope that helps to get you started on the road to recovery. Please ask if you need anything clarified or any further information.
The main Thyroid UK website also contains a lot of useful info. Just click on the relevant link on the menu on the left