Dude, when I was there, it was always more important to know the right people than to actually know the subject you were supposed to be working on.
If you don't like your LL6, I think you should start to make in-roads to another LL6 or another organization altogether if possible.
Keep in mind that when you apply for other roles using the internal system, your LL6 is immediately notified. So it will be in your best interest to cross that bridge with them directly before applying to anything. Spin it as a grown opportunity or something...
Good luck