Per the vet's advice, we have been tackwalking
Recently, we changed bits to the JP Korsteel Cooper loose ring with a copper lozange (that's a mouthful ... ba-dum-cha ...). I was hoping that this bit would encourage GP to hold his own bit, and I was hoping the loose feeling would encourage him to play with it instead of tilting his head and jiggling. So far, this seems to be working for him. Additionally, I've noticed that going back to the walk has helped to encourage him to lean into the contact (read: really take it, not just bounce off the pressure). It's also helped me to work on strengthening my reactions to his inattention.
In short, going back to basics is working well for us. Now, trotting tonight!