Boy, are you ever a tough sell! :) I'm lost except to suggest somehow obtaining an original copy of IBM OS/2,
http://en.wikipedia.org/wiki/OS/2
which was made around the same time and originally intended to run on the PS/2, and that does have a GUI interface (first included in 1988 with version 1.1) that you're looking for. IBM has canceled support for OS/2 lately anyway, so this may be impossible. Maybe stalk eBay for an original boxed set?
My only other guess - and this after you said 'no Linux' - is Tom's root-boot floppy, given in the source link. It's a Linux distro that fits on a 3.5-inch 1.44 MB floppy, and after that you can try to build up from there by compiling from source. But Tom's root-boot has no GUI, it's all command line.
You're running either an Intel 8086 or 286 at best, and probably have a hard drive around 40 MB or so, if it even has one. So even your modest expectations might be too much. My last idea is to prowl around retro-computing newgroups; maybe some BSD tinkerer will be interested in modifying an existing system to fit.
I'm quite a retro computing enthusiast myself, but wow, this one has even me wondering...