Interesting.
I just found an old CD I had, and found I could mount the CD-ROM drive in the Autoexec the same way we would the folders, and it worked fine, so I think maybe there's just something on the CD that's missing from the folders.
I'll try a few things and get back to y'all. If it only works with a physical disk or an .iso, I'll try a few things in that regard, too.

Anyone figured out how to get this game running yet where it is playable?

I can confirm, I'm trying to run this version of Carmageddon in Linux Mint 18.3 64bit and I'm getting the same black bar vertical lines problem as well as significant performance issues. I also confirmed that simply running the carma.exe from the Splat Pack folder also does a CD check, but you can get around that by running the 3DFX .BAT file, that seems to also skip the CD check.
Play On Linux install also failed, during the Wine install. :(
EDIT: Carmageddon 2 GOG installer worked with Wine!