I'm sorry to hear about your woes
Your BF could very well love you, but for him to say "cheating is because he doesn't like that you are overweight" means he's a tool, you need to DTB (Dump that Bast@(d). He's not an over achiever with you, he messed up, and even if you were thinner he would have done the same thing. He wouldn't be sorry he cheated, only that he got caught, and he would do it again in a heart beat.
I know it's hard, my mom is constantly on me about my weight, I think that she just likes to fight with me and she knows that is a weak point with me.
"I really don't see a way out, until I lose weight and am able to be completely independent. Being overweight has literally ruined me ): ***I'd give anything to be free of this situation***"
Your weight isn't holding you back, you believing it is, holds you back. If you really want something bad enough you will make it happen. Break up and move out of your BF's place. Become independent... Go back to school, you aren't there to be popular or make friends, you are there to get an education so you don't have to depend on your Tool of a BF or your parents. If they have a dorm system, move into the dorm so you are away from your parents and BF, and focus solely on school, don't try to get another BF or anything like that.
You wouldn't worry so much about what other people thought if you realized how seldom they do.
Those girls that your perceive making fun of you? Do well in school, get a good job and you will own them.
Good luck!