Shielded cables, baby ! I replaced three of the four cables connecting the target panels to the transponder board. It's virtually eliminated the problem. I say 'virtually' because I still get the very rare phantom hit. It's so intermittent that it is difficult to debug. I'll try converting the fourth cable routing the all cables differently.
I used a USB cable that I'd salvaged from an old webcam or something. It's thin, flexible, shielded, and the non-USB end was already terminated in standard pin sockets that can connect to the transponder.