Omg, all that sounds awful, I’m so sorry! Honestly the best thing would be to sit your mom down and really explain to her that this isn’t an aesthetic thing, it’s not like wanting a new shirt or haircut. It’s about your comfort, both physically and mentally. Your bras are simply unwearable. If your bras are hurting you, then there’s an issue. You have to be able to make her see reason, after all she has boobs too!
Check if there are any outlets in your area, and see when there are going to be sales. I don’t know where you live, but if you have a La Vie En Rose near you, they have really cheap bras (like $20 sometimes), which normally I wouldn’t advocate, but at this point it’s better than nothing and since you’re clearly still growing it almost doesn’t make sense to invest in the expensive ones.
Also, I know you’re young, and therefore not many of your friends will be well-endowed, but if you know anyone who might have hand-me-downs, hit them up. Something is better than nothing!