Earlier quoted context omitted.
Note that this does only the “show/hide’ the window part; The iTerm2 hotkey window, is a floating window, which for example also works in a space with another Fullscreen window/app opened (without moving to another space.)
You can get the rest of the way there by combining the Hammerspoon method of launching with WezTerm's ToggleAlwaysOnTop feature. I use a similar method described here: https://github.com/wez/wezterm/issues/1751#issuecomment-2299... You could probably even use Hammerspoon to initiate the keyboard shortcut if you didn't want to mark the window manually.
I expect that won't work on macOS, because an "Always On Top window" is not equal to a "floating window"...
It's described here: https://github.com/wez/wezterm/issues/1751#issuecomment-1973...