Well, I discovered one mess-up that I had to fix on the PCB design: the antenna junctures need to be as close as possible to their respective pins, and the traces need to be as short as possible. So trying to use the super-long trace on the FM antenna was apparently a bad idea.
I've moved the chip as far to the left on the board as I can manage it, as well as moved the regulating capacitor (C1) to as close to the chip as I could. In a future update I probably will completely rotate the chip 180 degrees to further shorten the distance between pins and antennas, though that'll be more involved than I want to worry about right now.
(I can tolerate cut-rate things if I'm making them for myself, but I do want others to be able to use this, so I'm going to make it as good as I know how to.)