The problem doesn't have anything to do with GSX, it's most likely caused by another add-on that, depending on the situation, releases or takes away the key you are using for GSX. Any add-on has the ability to intercept keys and not give them back ( or not ) at any time, at its choice.
And, CTRL+F2/F3 are not very good choices, because they are by default assigned to the propeller pitch.
If you really can't use the default ( which is fine if you don't have other add-ons ), try something with TAB, like TAB-F12 or Shift+TAB+F12. However, this won't save you if another add-on is using the ( wrong ) method of intercepting keys directly ( possibly using DirectInput or another Windows API ), instead of using Simconnect so, if it's using anything with F12, it might still cause a conflict.