_Auster_: If your system has something like Linux's unzip, you may want to extract the installer with it, then get the needed files (original game files, config files, etc.) and load them with the Rasp. Pi's DOSBox.
dtgreene: Unzip is installed, but how would I use it to extract the binary files from the .sh file?
The .sh file is a zip file with a prefix, just like most self-extracting zip files. (un)zip looks for the directory structure at the end of the file, so it will work with the .sh file just like a zip file, except it will warn you about unused garbage at the beginning of the file. The same is true of most self-extracting archive formats (rar, 7z, lzh being common, and I think the 7z command handles all of those, including zip). The only thing that might prevent you from using unzip with a .sh file is your shell. If it has smart completion, it often won't let you use tab on a .sh as the zip file argument. In that case, either type it out, use copy+paste, or do ^az^e to add a z to change unzip to zunzip, press tab for completion, and then ^a^d^e to delete the z. I guess. I use zsh.
I would suggest instead to install a version of innoextract that works with gog's Windows installers. That's mainly because of the DOS games I have, more than half don't have Linux installers, but they all work fine with Linux dosbox (although a few you might have to get glide and/or munt working with dosbox for the best experience on a rare few of them). I assume you have set up a cross-compile environment for building stuff yourself for your machine. If not, you probably should.
Actually, I('m not even sure why you don't want to just extract the game on your other machine(s) and create a tarball for the Pi. I mean, I understand your complaint about having to actually run the installer, but deleting the x86 dosbox binaries and using a bz2 tarball should save temporary space on your target (or you can use rsync/ssh or scp to avoid the tarball entirely).