X11's XKB (which, funnily enough, is also the standard keyboard layout system on Wayland) has support for Control, plus another 5 modifiers. Specifically, I believe Hyper is typically mod3. If you run xmodmap in a terminal under X, you'll see which keys are assigned to which modifiers. I have a custom XKB layout that assigns some keys to Hyper, and then adds Hyper to mod3. With X11 Emacs I didn't have to do anything to get it to work. But I don't think I've ever seen a pure GTK program recognize Hyper, so it may be a GTK limitation.

For learning XKB, I recommend the Unreliable Guide to XKB[0]. XKB is sorely underdocumented, so guides like these are the best way to learn it.

[0]: https://www.charvolant.org/doug/xkb/html/xkb.html

Please don't feel like you have to look into it, though. I was just wondering, since you mentioned that you used PGTK.