This is how sick care operates, their job is to keep you alive, not optimize health. Add to the fact the NHS is underfunded and doesn't have the funds for every man that walks in with low testosterone.
T3 only thyroid medicine, it's your only option. If you throw T4 medicine at the conversion problem, not much will change.
There is not much you can do about it, you have to play the hand you are dealt. More frequent injections has seen my SHBG increase 7 points.