The batteries arrived, and a quick test revealed why it didn't work with the CR battery: of course I swapped the terminals of the battery holder. Fortunately the voltage regulator I'm using has reverse voltage protection, so nothing burned. A quick desolder, removal of the plastic numbs that key the holder in the right position, and re-solder in opposite position fixed it.
Though I still need to compile the firmware for it.
I fixed the PCB, but I still want to rework it to use a QFN version of the chip — that should free some space on it too.