So, a little update:
I couldn't find the motherboard I was planning to use and had to do my testing on another - turned out that that one didn't support ISA option ROMs 🙁
However, I modified my assembly code to run as a DOS .COM file and got the CS chip properly initialized.
Adlib, digital sound, wavetable, Pi, GUS - all sound sources seemed to work, BUT the output sound was horrible; hissy, scratchy, garbled, etc.
I thought I might have made some kind of mistake in the analog circuitry, but it turned out that I'd used a 3.3V regulator instead of a 5V one for the "+5VA" rail. The CS chip was definitely not happy about that, and I don't blame it. Surprised it even worked at all.
I removed the regulator and patched a wire from VCC to +5VA as a simple test, and that did the trick. There's still a slight background noise when turning the volume all the way up, but that might improve if I use a proper 5V regulator instead - and besides, the output amp is probably way overpowered in any case for anything else than a high-impedance pair of headphones, so you wouldn't want to have the volume set that high.
Next steps are getting my hands on a more suitable motherboard and playing around with the option ROM + ordering a 5V regulator and testing with that.
All in all, pretty happy with the result so far! 😌