I've tired wrapping, with bandage, and tapes. And to be honest, all of them are unsafe. I actually binded in High School with Ace bandage and would just pull it as tight as I could get it because I wanted to be flat. To just like someone with a flat chest, or at least I could pass it off as I was injured. But I literally would be 2 seconds away from flat out bawling due to back pain. I've heard of using wet suit material to fashion a binder, and various other methods. The key to binding with any sort of tape is to never tape around your back, or very lightly around your back which means you'll never get it tight enough across the front to do good enough binding.
To me, the back acne, which I have rather badly only because of the binder, is something I can live with because of the plus of being flat. I'd say, to try out a large size binder. I wear an x-small tri-top, but it is actually just about the right chest size, and I can breath, work out, and function perfectly fine with it on. I'm a really small guy. I'm trying to build muscle now though.
But to be honest, I'd just try a different binder, a lighter grade or a bigger size. It has the least drawbacks and less chance to harm you in the long run. I know the acne sucks but with proper washing and changing when sweating out, you might be able to avoid it.