Quote: "GetImage + SaveImage"
GetImage can work but the OP asked about HTML so SaveImage won't.
Prime had demo'd HTML File Access HERE
(with Save) with some changes to AGKPlayer.js which may help.
meanwhile, OP also asked for the "entire screen with all elements
to the phone" implying the HTML canvas wasn't occupying the full screen of the device, so... basically, as i understand it, they want to access the OS "Screenshot" function from within AppGameKit and that's not going to happen any kind of "natively" as far as i know.
Quote: "click on the toolbar button (or press Alt+Shift+D combination)"
have those/buttons keys where you're from?