You have to shut the engine down before shifting the trans into park because the output shaft is spinning into the transfercase that is now in neutral. The transfercase in neutral disconnects the input shaft from the transmission - so it can and should go into park once you confirm the TC is indeed in neutral, and so that you don't have fluid-friction spinning of the transmission output shaft while towing - this could cause failure of the output shaft bearing if they are not self-lubricating. The manual is clear about this, however I'm not technically sure if there is an issue there or not, but I wouldn't chance it. Perhaps your mechanic can confirm. For many automatics, you cannot flat-tow them for this reason since the differential spins the transmission output shaft. Certainly even the Colorado/Canyon two-wheel drive cannot.
It is also a requirement to disconnect the battery as it is assumed the electric powersteering unit stays powered up so turning it with the front wheels while towing could damage it.
On mine I ran a fused lead from the battery into the cab to power a BrakeBuddy braking unit. And 4-wire trailer lead from the grill to the taillights and drilled for a set of 1157 bulb sockets.
This way nothing from the coach electrics touches the toad.
We want to make sure we get OP started out on the right foot.