Because apparently I cannot spend a Saturday like a normal person, I am now building my DOS machine inside OS/2 Warp instead of FreeDOS or DR-DOS.
OS/2 is actually good for this. I cannot tell what it is good for other than using it to run DOS sessions, but at least it is good for that one thing. As awesome as FreeDOS is, it still can't quite run Windows 3.1 properly yet, while OS/2 Warp does it out of the box.