I should have been clearer. I have no problem with performing the mapping, which I can do by using MIDI Parameter Menu (3.4.2.5) by entering the CC number with the keyboard. The problem is that the input of the mapped control doesn't have any effect on the mapped parameter.
I can map CCs 27, 28, 29, 32, 38, 43, 44, 52, 53, 64 to parameters, but incoming messages do nothing to the mapped control. In an empty MIDI mapping preset, MIDI CC 28 and 29 change the entire CS-80 preset. This isn't a hardware issue; I've replicated with Arturia MINILab mkII monitoring in Arturia Control Center, and using the CLI program sendmidi to send and CLI program receivemidi to monitor in a way that bypasses hardware. I suspect that 27 might be hardwired to "filter preset," and 28 and 29 are both mapped to some sort of change preset command, as I've seen similar functionality in Analog Lab, though I can't say I recall that those were the exact CCs.