375Worth: Game is installed, as an .sh file - whatever that is - but there's no option to do anything on the GOG screen that displays the game I purchased, the only thing to do is download that same file again. I am not used to 'no clear indication of how to use what you bought' games, apologies.
The .sh file is the Linux version of the installer. You have downloaded it with your web browser, not installed it.
I am not familiar with Chromebooks, but IF it works like PC Linux, then you would have to:
1. Open a terminal window (ie. command prompt, where you can type commands)
2. Not sure if needed, but maybe change to admin user (you can first try the rest without this step):
sudo su -
3. Go to the directory where you downloaded the installer, e.g.
cd /home/375Worth/Downloads/ (replace "375Worth" with your ChromeBook username)
4. Make that .sh file executable, as most probably it is not executable after downloading it to your Chromebook:
chmod +x thefile.sh (replace "thefile.sh" with the actual name of the .sh file).
5. Run the installer:
./thefile.sh
Anyway, I don't have high hopes the game will run on your Chromebook, even it it had a x86 CPU. The game requirements state OS as either Windows, Ubuntu Linux or MacOS X, and ChromeBook doesn't run any of those.
Also, the game states it requires graphics processor "nVidia GeForce GTX 660 2GB / AMD Radeon HD 7850 2GB" (or better), which your ChromeBook most probably doesn't have either.
So, yeah, you bought the game for a wrong system. Your ChromeBook is probably not fit to play so called "PC games" (either Windows or Linux versions of x86-compatible games), it is probably meant to play Android games downloaded from Google Play Store only. So yeah, get a refund.
Also what you said about the game not warning you about being incompatible with your system while you downloaded it, it was just a normal web browser file download. Your Chromebook or the web browser does not have any way to check whether the file you are downloading is somehow "compatible" with your Chromebook. It is just a file.
If you tried to execute that file on your Chromebook, THEN you would quite fast know whether the installer is even supposed to run on your Chromebook. (I presume it is not supposed to work there, or at least it is not supported.)