This is a DOSBox game, no? You likely need to run from inside DOSBox.

That sounds logical.
Sadly I don't know how to apply DOSbox to a program.

I don't own this personally. First, I would double check GOG doesn't include a shortcut (Start Menu/Game folder) for this already (they quite possibly do not). Second, if it was me I would copy/paste the existing shortcut and rename it to include FACE in the name. Copy/paste the _single.conf file to face.conf, edit the file to run FACE.EXE instead of the game EXE, and lastly edit the pasted shortcut's target to use face.conf instead of _single.conf for the second -conf listed. That should give you a permanent way of launching Face Paint.
Alternatively, you can backup the _single.conf, comment out the last few lines (so it only mounts C) and edit the Windows shortcut to remove -exit if needed.

Directory listings/screenshots will allow me to be specific it that sounds needlessly complicated. It isn't really.

I have the game on Steam and it has a shortcut to run the face paint program (which uses a file called dosbox_cotfr_face_windows.conf). As Gydion suggests, make sure there isn't already a shortcut. It would be surprising if the GOG version was missing this.

In GOG Galaxy you can run the application here (screenshot)
Works just fine for me. It is practically a little DOS paint program that allows you to paint yourself or edit the stock images

In that case the GOG Galaxy solution sounds the easiest to me! :D
Thank you all for your help!