I finally got the Programmable Interval Timer to work with interrupts so I have a proper clock. Dear goodness that took a while. I still need to make it a bit more accurate and actually check that it's working correctly.
Before I just couldn't figure out why it sometimes works and sometimes doesn't. I started digging through Allegro's source code to see what it does and there was a note that "sometimes the timer just doesn't register so we loop the timer set-up function four times." That actually fixed it.
I also started to create an implementation of Mode7 for fun but unfortunately even at its best it runs way too slow. I'll have to experiment with that some other time in the future.
Edit: Made a build for DOS/4GW and my program suddenly runs a lot better. Mostly the time it takes to clear and copy the framebuffer into the VGA memory cut by a lot. I think I'll support both 16-bit and 32-bit DOS.
Edit2: Tried to get Mode-X to work. No luck yet. I'm going to have to rethink the way I do my rendering for it. I think I'll return to it another time.