Couldn't you have a batch script which overrides windows environment variables?

Something like this:
https://superuser.com/questions/1226025/how-to-redirect-appdata-roaming-to-a-subdirectory-to-make-an-application-more

It seems like too much work to me and probably not something worth risking your job over but it seems like it is possible (there's always the slight possibility that some games might not respect the env variables and write to what it assumes is the right place...)

I'd suggest emulation. NES/SNES/Genesis are small enough games, and don't take that much processing, games ranging from 100k to 8Mb in size, and the emulators will be small too. No installation required.

Other emulation you might consider is C64, Atari800, and Apple 2, sorta like DosBox games but slower. Probably a number of text adventure games as well.

Next option up from that would be games you can extract as a zip and run immediately. Quite a few i can see likely in Humble Bundle which are just zips, while GoG games all want to be installed.

Up from there, you might copy gog games and try them on another computer and see if they work. If they don't have any difficult dependencies or require keys in the registry they should work fine too.