Update: I still have an issue, but have seen some behavior that might offer a clue.
The motherboard I’m having trouble with is a new-ish MSI MS-7A72. On the chance that it was a BIOS issue I downloaded and flashed the latest version (currently E7A72IMS.160, dated 7/11/2017).
That did not fix the problem. But the system seemed otherwise flaky after the update, so I decided to clear the BIOS NOVRAM using the battery jumper on the PCB.
The first boot after resetting the NOVRAM I could see the PCIe Lime board. But on subsequent boots it was no longer visible. This behavior is quite repeatable.
Given this clue I was all over the BIOS settings, but have not yet come up with any combination that lets me see the board. The only way I can get it to work is by resetting NOVRAM.
Does this suggest anything to anyone?