All I can say is, if you think an LP feels unbalanced, don't even bother trying an SG, especially standing.
I don't believe that either. I know that's story with SGs, but I have three of them and none of them feel unbalanced to me. They just hang there like they're supposed to. There's no neck diving or anything. But I also play as low as a strap will go. Maybe that somehow counters the imbalance.