I gave it a few pokes at some point but turns out that menus on Windows are tied to threads so you can’t just cheat this by calling the menu functions on a separate thread and grabbing the results when done - you have to update/render the game on a separate thread, which isn’t currently possible in GameMaker unless you have an Enterprise subscription for source code access and a lot of time on your hands. There might be undocumented tricks that I’m not aware of, it’s hard to tell with WinAPI.
If you just bought the extension and for this specifically, you could request a refund and buy Windows’ windows instead - a custom window is how all major web browsers do context menus these days (you can see it with AutoHotKey’s Window Spy).