The third Live CD I tried, worked. Puppy Linux 5.4.3, "Precise Puppy", based on Ubuntu. I had to download the ISO, burn a CD, then install to a USB drive (Setup > Puppy Universal Installer). Once I booted the USB drive, I installed the following packages using Puppy Package Manager:
libSDL_image-1.2.so.0 => not found
libSDL_mixer-1.2.so.0 => not found
libSDL_ttf-2.0.so.0 => not found
(From ldd of the ufo executable.) That's enough to get the Desura copy of UFO:AI to run on my Intel DH67CF system. Whew.