The above is excellent advice.
I'd add that the best way to show him that you're mature enough to make that decision is to be mature in other ways: be responsible about schoolwork; your job if you have one, and about money generally; be thoughtful and helpful toward other people; and don't be a jerk in your social life.
There may not be anything you can
tell him that will change his mind, but if you can
show him that you're capable of making good decisions in other aspects of your life, it'll be easier for him to see that you can make good decisions about this.
In the meantime, a mature way to handle this problem would be to sit down with him and (calmly) explain that this is really hard on you, and ask him to help you deal with it in other ways: for example, by letting you get your name and gender legally changed, and by supporting you with school stuff. (Wait to do this until you're sure you can pull it off without getting too emotional or having the conversation turn into some sort of fight!)
It sounds like your mom is pretty supportive -- you could talk with her first, and ask her to help you come up with some ideas along those lines.
Also, a big part of dealing with this stuff is to get some perspective on it yourself: recognize that it is a huge, life-changing decision, and that waiting, while painful and difficult, isn't the worst thing in the world -- you know you
will be able to do this in the foreseeable future, which is a heck of a lot better than not seeing any way at all to do it, or not even knowing that it's possible.
Nothing says childish like jumping up and down and screaming "But I want it NOW!!"
(I kid, I kid -- I know that's not what you're doing, but on some level, that may be what your parents perceive. It can be really hard, and painful in some ways, for parents to see that their child is grown up.)
