Go for it man, you have the right attitude. If all else fails I just wanted to suggest contacting Portland's LGBT center or the closest LGBT center near you. Mine does $20 therapy sessions a couple days a week that might could talk to your aunt and uncle if they have that or you could take them to a transgender meeting, hearing others stories might make them realize how serious it is.
Come at her logically. Just remember to keep your cool. Try researching statistics that show there aren't adverse effects for younger kids. Also show them some cases where people under 18 have gotten on T, I know there are a few. Tell her you are transgender, you've made your decision, the research of others getting T under 18, and then again say you don't understand why you should have to continue being depressed when it's only a little bit of time before you're 18. That's what I would say. You know her best though so you probably know the best approach.