I've played with the same idea, although my hard wired and haven't wired in the the heaters up yet.
Not sure if the MKIII has mirror toggle switches on both sides, but on MKII on the drivers side I took
and ingnition live to feed to the electric mirrors and took a connector off a door loom with a mirror
connector off and run the other 3 wires from the drivers door through the loom into the cabin then
into the passenger side alot on effort but pretty pleased next is to wire up the heater control with
the dash switch. I originally wanted a temp reading off the passenger side mirror but i believe that
will only be possible with a UCH upgrade or something along those lines.
With some of these ...,,..,
Just wired it through trial an error and testing what each wire does on the electric mirror connector.