The way I’d have expected to do that would be to override the globalToolBarItem of the Kirigami.Page, but it’s read only, so I don’t really know how to help you with that. You should ask in the Kirigami Matrix room or in the Discuss forum.
Thank you very much for the detailed answer! So there is no “quick and easy” way to do it :(
I’ll try to look for info in the matrix room you linked, thanks!