I think the first question to answer is whether booting from a game floppy image via the Gotek is the best way to achieve game compatibility with an expanded Amiga. In my experience, compatibility is better when using WHDLoad images instead of unmodified floppy images for expanded Amigas.
You mentioned you have a 4GB IDE-CF HDD. If you already have WHDLoad installed, what has been your experience trying to run games through that route, with and without your Hawk expansion unit?
If WHDLoad has not been tried with the Hawk, I recommend trying this first. If you get good results, then your problem is solved. If the Hawk still give you problems, I would try swapping out the memory on the Hawk.
As far as the recommendation to run "SetPatch" in your link, that may be important too. And this is another reason you should be booting from the CF card into a properly patched environment for your expansion hardware. WHDLoad will support launching games from such a patched environment, and this is one of the reasons it provides for better compatibility with expanded Amigas.