I finally figured it out, and in case anyone needs to know for future reference:

Option 1: Go into "Slot Devices" by pressing TAB to get into the MAME config menu and choose "Joy" under "ctrl2" - This is not set by default.

Option 2: Add -ctrl2 joy to your command line in emulators.cfg - This will force a controller to put put into controller port 2. So for example, my command line for Astrocade looks like this:

lr-mess-astrocade = "/home/pi/RetroPie-Setup/scriptmodules/run_mess.sh /opt/retropie/emulators/retroarch/bin/retroarch /opt/retropie/libretrocores/lr-mess/mess_libretro.so /opt/retropie/configs/astrocade/retroarch.cfg astrocde /home/pi/RetroPie/BIOS -ctrl2 joy -cart %ROM%"

@SpudsMcToole Oh, if it helps - when I get the MESS copyright splash screens and it says "Type OK or move joystick left then right to confirm", the joystick is recognised and works but the keyboard doesn't. (The keyboard works fine with other emulators.)