Gamemode menu won't open

Not too sure what you mean by "gamemode menu". There is, however, a shortcut to switch between creative and spectator mode, which is F3 + N. In your case, it would be Fn + F3 + N.

There's also F3 + F4 in the 1.16 pre-releases that is supposed to bring up a gamemode menu.

What operating system do you use? Do other keyboard combinations like Fn + F3 + G or Fn + F3 + Q work?

Cannot reproduce, pressing F# + F4 at the same time opens the menu properly

I cannot use the gamemode switcher for some odd reason. All the other keyboard shortcuts work, however.

@Ethan Lessard, try using FN + F3 + F4.

Also try using a third party keyboard, some laptop keyboards have issues with the game.