If male chest aesthetics weren't important to you (ie, you just want to have a flat chest but you wouldn't be upset if you couldn't go swimming shirtless), I'd start medically laying the groundwork for a reduction. Insurance will definitely cover a reduction. Go to your doctor, tell them about your back issues. Don't mention binding, don't bring up any gender issues. Just stick with the "my back really hurts, it has for X amount of time, I've tried XYZ to make it better but nothing helps".
Reduction probably won't be a doctor's immediate go-to. I have a friend with a fairly generous chest and prior to getting it reduced, she was prescribed painkillers and went to 12 weeks of physical therapy. These were all covered by her insurance, but she still had to wait almost 6 months from mentioning it to her doctor to getting the actual reduction.
If the aesthetics matter to you, there are options other than insurance. Lots of people take out loans to cover medical expenses. If you have a job and would be able to pay the loan off over time, it would be a way to pay for it and build your credit at the same time.