Hey, trans guy here. I have read through this thread and noticed people seem to be sending a lot of misinformation around.
First of all, binders are not as dangerous as people are making them out to be. Yes, they can damage your ribs but this is typically dependant on the company you buy from (avoid cheap ones off amazon and anything that uses clips.) You best option is to buy one from GC2B or Underworks. Make sure you get the right size and don't let them wear it for more than 8 hours per day (encourage them to take breaks if they're in pain or struggling to breath).
Putting them on the pill does is a good idea, it can't really do any lasting damage. Lots of ftm (female to male transgender) people I know find periods to be really difficult so it should help them a lot.
As for the whole transitioning side of things, encourage your child to take it slow. It's so frustrating not being able to transition, I'm in an environment where I can't be out to the ones around me cause I don't know if I'll be kicked out or mistreated so I can understand why they want to start the medical side of transitioning as soon as possible. This isn't a great idea though as most of it is irreversible so you should make sure your child is 100% ready before they start hormones or get any surgeries (though I don't believe surgery is available to under 18s).
The best thing to do is let them socially transition and make them attend gender therapy. Social transition involves letting your kid live as the gender they identify as, going by their preferred name and pronouns to make sure they're comfortable with it (I believe this at the very least used to be a requirement for medically transitioning under NHS). Gender therapy is like regular therapy accept about gender, it gives people a safe space to talk about their gender identity without judgement with someone who understands what they're going through.
You said you were struggling from it and I can understand why, this is a huge thing to put a yourself through. If you need support, attend therapy. It will give you a safe space to vent about how it's impacting you and will hopefully make this far less stressful. But don't forget that what your child is going through is horrible.
Gender dysphoria made me suicidal and depressed, for years I felt like I'd never be happy because I was always trying to be a "girl" like society wanted me to be. When I came out to mum at 15 she yelled at me and called me a waste of money and said I'd never be her son which hurt like Hell and has had a huge impact on my mental health. I'm nearly 17 and I feel like I'm only just starting to find myself. Transitioning is a scary road and your child needs your support and love more than ever.
It's clear you love your child and it's commendable how much you're supporting them, it's also very normal to find this hard and scary. Try and find parents of trans kids online or watch trans youtubers (I'll link some good videos by trans youtubers at the bottom of this post.)
I don't know how helpful this comment is (I mean this thread is really old) but if any parents of trans kids happen to be reading this I hope it helped you in someway :)
Helpful Links:
(made by a trans man called Miles and features stories from lgbt parents)
(another video by Miles, shows his transitioning experience during his first year on testosterone)
(video by a trans man named Sam in which he talks with his mum about his transition)
(video showing how you can support your trans child and how it will help them)
(video by trans women named London who made and Q&A video with her mum)
www.gc2b.co/ (GC2B website)
www.underworks.com/ (Underworks website)