Unfortunately is not possible to have an hotkey to start the GSX menu, because of limitations in the SDK, the manual explains these issues at Pages 66-67 and, in a nutshell, it all boils down to the removal of two crucial features in the SDK which has been there since FSX, the Simconnect menu and the Simconnect short text, which are in fact the same API call.
In P3D, GSX was waiting for its hotkey (which in MSFS cannot even be reserved exclusively, another missing feature from FSX), and when it receives, it conjure a menu on the fly.
In MSFS is the opposite: we can't create menus on the fly, so it's the menu that must call GSX instead.
This is just the basic concept, in reality is far more complex because, to overcome the reduced communication capabilities between the Html/JS menu system and Simconnect, we had to use a very convoluted cooperation between Javascript, a WASM module and Simconnect, just to replace the missing menu functionality.
It could be so much simpler, so much more responsive and easy to do, if only Asobo/MS added the ability to call the HTML/JS directly from Simconnect ( through Coherent ), we asked for this long ago, but it doesn't seem to be much interest, we can only hope GSX popularity might push things forward.
Anything to control the menu system?
The upside of all the above is, now GSX is inherently ready to be controlled remotely.
But we want to make a proper interface, so we want to discuss first with premier airplane developers, hear their suggestions, because the idea is, with a proper integration, you might not even need the Toolbar menu anymore...