The "TOS emulator" runs fine, so I pulled out the soldering iron and installed the STE "DMA fix" I also bought.
This is a recent-ish hardware solution to work around a signal timing issue with the older C025913-38 DMA chip in these machines, that can cause data corruption with ACSI peripherals. Atari made a newer version of the chip, the C398739-001A, that doesn't suffer from the issue, but that's hard to find. This adapter board offers a solution for the older chip.