I decided to abandon using both the PCI-based IDE HDD controller card, and also the ISA-based IDE CD-ROM controller card. Instead, I plugged an old HDD directly on to the mobo's onboard IDE controller, using an old 40 pin cable. I connected the cable's middle master connector to the HDD, and also set the HDD's jumper to master. Then, I connected the ATAPI CD-ROM drive to the end of this cable, using the end slave connector. I also set the CD-ROM's jumper to slave setting. This all works.
It's interesting, because when "pure DOS" detects the CD-ROM drive, it states: "Programmed I/O" mode. I guess the mobo's BIOS only understands this mode, and not UDMA? This could be why my previous tests were failing. In these tests, I had connected the CD-ROM drive to a 16-bit ISA-based CD-ROM controller. My guess is that this controller card was trying to use UDMA mode, and the mobo wasn't happy about it. Just a guess.
Edit: Actually, I'm not sure if my guess above is right. Perhaps this test failed for another reason. I notice the mobo only has 1 onboard IDE header (because the other header is a SCSI connection). Maybe on this specific mobo, you can only have 1 master IDE device, and 1 slave IDE device? My previously unsuccessful test had 1 master IDE HDD, and also 1 master IDE CD-ROM drive. I'll retry this test, but set the IDE CD-ROM drive to slave, even though I'll plug it in to its own ISA-based CD-ROM controller.
Edit 2: I think I was right! (Wow, that's a first!) I set the CD-ROM device to slave, even though it was plugged in to its own controller, on its own, and effectively had no master device. It worked. "Pure DOS" detected the CD-ROM device, and stated "Programmed I/O" as before, for its mode of operation.