I uninstalled the tuner module during a gas stop, and the problem went away. Unfortunately, so did all the power to the player, which apparantly was somehow wired through the tuner. When I plugged the tuner back in, both the power (:)) and the problem (:() came back.

I'll bet this is some kind of a ground loop going through the tuner. Your player is getting its ground through the antenna wire connected to the tuner instead of getting its ground from the ground wire on the sled.

The only way I can see that your player is getting random button presses is if there's noise on the serial and/or stalk inputs. I haven't scrolled back in the thread, but did you try disabling the stalk in the player configuration?