Nice smoker! I think they changed the gp's on the 1782 to take the full 12 vdc by looking at the schematic.
The 1772 and prior built had the resistor in series with the gp's dropping the voltage to the plugs.
It still wouldn't be a bad idea to use a solenoid to keep all that current out of the key switch. Just use the orange wire that attaches to your gp's now to energize the solenoid.
Don't buy any Kubota parts from Cub Cadet. I get all my Kubota parts from Messicks.
http://www.messicks.com/ Get the part number from PartsTree and then take the "KB" off of it and use that number on the Messicks search.
Oh ya, we like pictures so keep us posted of your progress.