[bestgenman]

I'll add something else to the mix. The solenoid that holds the fuel rack in the run position is a two winding device. One winding is pull in and the other is hold in, the hold in is fused. The pull in is tied to the starter so that when the starter is engaged the rack is pulled into the run position. The hold in is connected to the IGN circuit of the engine which includes the alternator excite, engine gauges, lift pump, & other accessories.

It sounds like your IGN circuit is not at +12 when the engine starts. check for 12VDC at the hold in of the solenoid (red wire). I am assuming your engine has this style of solenoid as I am not familiar with the 12.5CCO Kohler. This is a common compliant for this scenario.

Your transfer switch appears to be manually controlled by the switch on the dash. some are two position and some are three position (shore-off-gen).

If you have a squirrel cage fan in a side bay, it forces air into the generator compartment and exiting through the radiator to the outside. If you leave the generator compartment open the radiator does not cool. This fan is connected to the AC output of the generator ahead of everything. There are a number of other schemes such as sail switches etc.
I hope I understand your question!!

Gordon