Piece of cake!
Not here to teach you about your grill or how to solder (I would fail miserably at both), but here's what I did:
I added an aircraft style switch to my Pitboss Lexington. Pictured below. First pic is of the circuit board with the bare "Prime" switch highlighted. Second pic shows the wires soldered to that switch for the mod. I figured out the right poles to use with a multimeter. The default between these two poles is OPEN. When you push the Prime button, the resistance drops to zero consistently. With the switch wired to these two poles, I can start the auger motor with the Prime button OR with the switch that I installed. When I use the switch, the auger motor simply runs unattended.
So nice for priming before a smoke and clearing the auger after a smoke!
I just ops checked it for the first time. Works great and I am about to mount the switch right under the prime button on my grill and button it up for my next smoke. I'm so happy!
You could do this with any on/off switch. I REALLY like the idea (now!) of using a timer like you are showing. But I had one of these red cover switches on the shelf, so I'm using that for now. Now that the wiring is in place, I could easily use just about anything.