Hi Brittani, I wish I had the perfect answer for you but I don't. I think you can take peoples ideas and then pick the one or ones you like.
Your idea about explaining to them is a good one and it might be better if you mentions some of the numorous studies that have been done recently that shows our brains have a gender identy and tell them you were born with the brain of a female and now per doctors orders you are changing the outside to match the inside

I think if you hit them with some of the technical stuff it will make them feel small if they are closed minded and it they are open minded it will make them understand. Thats one of my ideas, I am sure others will come in and give there ideas too.
I just thaught of another one, lets say you 28 years old then all you do is look at them with confidence and say I am her daughter, what's wrong have'nt you ever seen a 28 year old girl going through puberty or just say I'm her daughter thank you, I was a little late in starting puberty.
Don't worrie about how they act, just tell them you are her daughter, her trans daughter if you want to and if they give you a hard time tell them you are suposed to be professions and we are here for my dear mother let us stay focused on that so she gets the best care possible.
Good luck and aloha.
Jennie