Sharky, I'm FtM and going into law enforcement. It's true stealth isn't really an option, but thus far it hasn't been a big deal for me. I listed my gender on my application as "M" and my birth name as prior legal name. When I went in for the first interview they saw my old name, asked if it was a birth name and for the documentation of the name change. They have never mentioned gender and I'm optimistic about my chances. There is even a department in Vermont that lists "transgender" as an option for gender.
Have you thought about looking into TCOPS (Transgender Community of Police and Sheriffs). Its a professional organization for transgendered officers. They also have a mentoring group on yahoo that I'm a part of. They've been a great help to me during the application process and reaffirmed my faith that even though I'm trans I can still work in the field I love. Their website is:
http://www.tcops-international.org/ .
I also wouldn't worry about your height. I'm only 5'3" and I've known several officers (cismen & ciswomen) that are shorter. One of the officers I road with joked that for "guys like us" the 5' wall is harder because it's our height. With a bit of training though it is possible to get over it and I'm looking forward to my physical agility test tomorrow. Size isn't a deciding factor in law enforcement, command presence is. Being able to portray confidence and control when staring at someone twice your size will serve you far better than an additional half-foot in height.
For the quick answers to your questions:
How did you pick your major/career?: A ride-along a year ago introduced me to patrol work. 250 hrs of ride-alongs later and I know there is no going back. I added a major in Crime and Delinquency Studies and started physical training. Now I'm in the application process.
Is it something that's typically very masculine?
Traditionally yes, police work is seen as masculine. Last estimate I heard was that 1 in 10 officers are female. That's not why I'm interested in it though. I'm actually drawn to the more "feminine" emotional connection and working with people who need help. I like police work because it provides a good balance for my instinct to "mother" and help as well as my drive to protect and study how to handle violent encounters.
Do you feel being trans is holding you back?
Partially. I'm applying for larger agencies than I ordinarily would because I expect their going to be more accepting. I did have one department that I suspect my trans status was part of the reason I wasn't selected to continue the hiring process, but that could also be because I transitioned during that application. I've spoken with trans officers and know that it is possible to get hired, just a little harder than if I was cis.
I hope this helps. I'd follow your dreams. To me giving up and letting society force me into a career than what I want would be just as bad as not transitioning. YMMV, good luck with your decision.