I disconnected the neck, cleaned the contacts with a cotton swab and alcohol, found that the contact plates are worn in spots.

I haven't played guitar often until now, so I've go no idea how it can be worn out. I used a folded piece of paper underneath the neck to change where the contact points connect to the pins, and that has fixed it. I also borrowed a set of Torx drivers with the T8 and T10 sizes, so I may also go ahead and move the PCB further forward on the neck, which is another recommended fix.
Now that I'm trying to learn these new songs, meet some of the gold/platinum/diamond Challenges for additional in-game unlocks, I'm more frustrated that GH3 has such a cheap build quality. Then again, one of the buttons is off on my Rock Band guitar, so it's not like Harmonix has the higher ground.