I would be really surprised if a double dose of desogestrel taken for longer eg 3 months or more wouldn't shut you down. This would be my first go to if tolerated.
Secondly, and I can understand you not wanting to stick your head above the parapet if you are receiving a therapeutic dose of estrogen and testosterone through the NHS, but you could pay for a private ultrasound to see what is going on.
I don't know if an IUS is an option for you but that is generally good for getting rid of bleeding.