Untested partial solution that you may already have tried:
In the window manager's keyboard settings, create keybinds for raising and lowering windows.
Create a script that uses dotool, a third party tool which can send keyboard events and mouse movements, to call the previously configured keybinds.
Missing bit: Figuring out whether the window is raised or lowered to know which keybind to send.
The author of dotool says that they wrote it because ydotool (the alleged successor to xdotool, I assume), needs root and a background daemon. That said, the linked page seems to indicate that dotool also needs some permissions.