Since you didn't mention this, I guess you don't have proof that the PB admin actually said that to you? That would be the best proof.
I don't know of any way to bypass a PB Global Ban because otherwise, cheat users would be paying for that instead of another Battlefield 3 key.
As a last resort, see the admin who banned you can also unban you. Talk it out with him because by now, he's probably not angry about not getting his LAV seat. Don't say "Hey man, just because you didn't get to sit in the LAV seat you wanted doesn't mean you should abuse your power." Do not mention you miss playing Battlefield. He might be sadistic and enjoy your anguish.
Pretend you're young or hurt and add emoticons. Lie if you want because obviously his morals are a little loose and once you have your account back, you can teach him about his loose morals. Maybe add a story for dramatic effect of how you're undergoing opiate treatment addiction and being in the third week of therapy without opiates is making you angrier than usual or something.