The neutral starrter switch is mounted on the side of the shifter, not on the transmission so it could still be your problem. The other thing that could be happening is the Protective Control box could have a bad solenoid in it. There is a large solenoid that looks like a starter solenoid that's used to detect when the batterys have been installed properly to prevent damage from batterys being installed backwards. It could be dropping out when it gets warm and them remaking after it cools down.
The only real way to test it is to exchange the PCB with one you know is good.
The problem could certainly be in the switch but they usually don't act that way. If you have a meter you can chase some signals on the STE-ICE connector plug to see if the start signal is getting to the motor or you can have someone go under the truck while you turn the switch to see if there is 24 volts getting there.
FTM2 USS Long Beach CG(N)-9
US Navy, 1969-1975
1986 M996A1 HMMWV ambulance w/ Kasel M101A3
1977 M880 Dodge Crew Cab
1966 M274A2 BMY Mule
1965 M151A1 Mutt w/ Stevens M416B1