It looks like a crazy ball of string and rubber bands now. I added an external SSD in an enclosure powered by the compatible JMS578 chipset. But the board itself cannot supply enough power through USB to external drivers, so there’s also a powered USB hub (thus the whole thing needs two power supplies).
It works is the best I can say about it at this point.
Important edit: I discovered that the powered USB hub is not necessary (presumably because this is an SSD, not a spinning disk). That eliminates the power supply problem.