Trans Without Politics

Hi, I'm new here (and to all forums) but I am looking for advice and this seemed like possibly the right topic to post it? I'm FTM trans and I'm a medical student and currently really struggling with the career thing whilst trying to transition. I'm British and work in the NHS and i'm on placement all the time now which means I'm in hospital on ward rounds etc. and I'm not on T yet, been on the waiting list for treatment for about 6 months now and it's gonna be about a year longer unless I can get up the money to go private. So i have to wear an ID badge which unlike most jobs needs to be my legal birth name and is round my neck all day, i pass about 50% of the time but I'm really struggling with whether i should come out to people there. I'm out to my friends and family but I don't know how to cope when every day I see a ton of different patients who are already often in a state of confusion and then they think i'm male, then see my name badge, then get confused, and I don't know what to say, especially in front of the other students I'm not out to. I feel like I'm lying the whole time and i feel **** every time i have to mis-gender myself and sometimes I just want to give up and take a year out to transition but I know that would really hurt my career. I don't know anybody else in my position, I've never met any trans people let alone in healthcare, and I just don't know what to do, do I come out to the medical school even though it probably wouldn't help me day to day? Has anybody else had a similar experience at work  or has advice generally about coming out there? Somtimes i just regret so much not coming to terms with being trans earlier, years of supression here... Thanks