One side of the load contacts on the remote start solenoid is hot all day. It comes off the starter hot cable. Other side starts the bendix on starter.
On all coaches I've spelunked, there switched side of the chassis battery disconnect switch would make a good alternate source of 12V, and an inline 30A fuse would be a good addition. I put a 50A switch in line on that hot leg.
Per Bowers, Cole Hersey 24059 solenoid ($20 on Amazon) is a good quality replacement. Wouldn't hurt on coaches w/75k on the clock or more to replace the original for PM purposes & carry the original for a spare.
Now if you turn the key switch and the motor doesn't crank, you can jump 12V from the perma-hot load stud to the small white wire stud (make sure you are in Neutral, have park brake ON, you are not in the path of tire travel, etc.)
Baja-tested '08 2-slide 36'
Alpine: The Ultimate DIY'er Project