After a little experimenting, I found the right options to use. I had thought it needed a USB-specific device, since "arduino" is configured as a legacy serial device while "arduino-ft232r" seemed more appropriate for this TTL-232R cable, but here is what actually worked:
avrdude -c arduino -P /dev/ttyUSB0 -p m328p -Uflash:w:atmegatron_2.4/AtmegatronV2-4.hex:i
I also had to experiment a little to discover that the atmegatron actually needs to be connected to its normal power supply in order to flash. On my other devices, the only power at flash time comes from the flashing cable itself.
No special drivers or anything are needed. Just avrdude. This was in debian/testing with "apt install avrdude". I may also have other relevant stuff installed since I use other little Atmel chips regularly, but I don't think any of the other packages are relevant for merely swapping firmware on the atmegatron.
To install factory presets, avrdude has other options to read/write eeprom. I'll figure that out later.