I have read many similar instances on the forums, and note that you suggested downloading COUATL 3 Public Test, which I did, but I still cannot access the menu.
This suggestion is relevant ONLY for users that don't have the Couatl powered menu, usually because of a conflict caused by other addons and/or missing system libraries. Your case seems different.
I have tried changing my key mapping to other values, but to no avail.
If you mean you used the "Key Mapping" menu the "Couatl powered products", it means you HAVE the Couatl menu, which means the software is running correctly, and I assume you DO have GSX as a product that can have its keys remapped in the Key Mapping menu, which indicates GSX is installed.
So, the only possible explanations left are:
1) Another addon is using the same keys assigned to GSX, so you must change them in the other addon, or change them in GSX. Some addons don't correctly handle the ctrl/shift keys so, if they use F12 (for example) they prevent you to use anything with F12.
OR
2) GSX is not working, but cannot call GSX on a parking for some reason, for example the airplane being too big for it. Usually, GSX would warn you with an error message and a sound. But if you put the "Message Verbosity" slider in the GSX Settings all the way to the left, because you didn't like the messages, you wouldn't see such warning.