That's strange. I feel that if it was really the fault of the IDE controller, Windows wouldn't want to install/boot up at all. The reason why I say this is because for things like SCSI cards that Windows doesn't have drivers for, you need to install the drivers for them during setup to be able to use the HDD (i.e. in Windows NT4 setup, you would manually have to insert a driver disk after it tried detecting any controllers so that Windows could be installed). I'm not sure how Windows 98 handles things like that, however.
What make/model is the CD-ROM drive? You might be able to try temporarily installing something like Windows 2000 or XP to see if the CD drive shows up in those versions. An even better option might be to try and install the CD drive into another vintage system (if you have one), and install Windows 98 on that, and see if it shows up there.
Another thing you can do is switch up the positioning of the drive. If it's the master drive of either the primary or secondary IDE port, you could try switching it to the other port, or try making it a slave drive to the port that the HDD is on, and configuring the HDD to be the master drive. As a general rule of thumb, I always make the HDD the primary master drive, and use the CD drive as a secondary master. Sometimes, if the jumpers on the drive are wrong, the drive won't even show up on BIOS setup (for example, if your drive is the only drive on that port, and you have it jumpered to master). I've been fooled by this a couple of times and have had no idea what was going on until I looked into it. This might be a case similar to yours, albeit, it shows up in BIOS and works, but doesn't work in Windows.
You might need to try and get multiple versions of the chipset driver online and go through each one until you find something that works. That's all I can think of at the moment.