I figured it out, at least in Linux Mint:

1. Install Keysharp (which includes installing Microsoft’s .NET 10 Runtime, but not SDK!)
2. Put this in a .ks folder:

!s::{ ; Press Alt+S to use sudo on the previous Terminal line
    KeyWait 's'
    If WinActive('ahk_exe gnome-terminal-server')
        Send 'sudo {!}{!}{Enter}'
}

Set it as a Startup Program with the command keysharp "(filepath)/(filename).ks" and it’ll always be ready to go!