My 3316 CAT will start right up when the engine is cold. Once it warms up, if I shut it off I get nothing but a click under the dash and dash lights all light up. I then have to wait 1/2 an hour or more before it will start again. Sometimes if I haven't waited long enough I'll turn the key and nothing will happen at first but if I hold the key in the start position after a few seconds it fires up but if it is really cold it fires as soon as the key is turned. Anyone experience this?
I'm thinking your main starter solenoid (heavy solenoid that is built right on the side of the starter) is sticking when hot. If possible, with a helper - and if it can be done safely - have the helper carefully tap on the starter solenoid when having the problem and trying to start. If tapping makes it start cranking, you probably need a new starter, or a starter rebuild.
As Peter stated solenoid issue. Commonly caused by "heat sink" Chevy Corvettes when new had the problem they called hot start issue. They solved it by putting a remote solenoid away from the engine. My "81 FT would sometimes do the same thing I would crawl under and jump the battery cable "in" to the starter motor "out" post on the solenoid and it would then start.
You're a braver man than I am... ^.^d
It had a full frame no air bags plenty of clearance, not like my '99
Brakes on and in neutral (shifting cable not electronic) usually happened when stopping for fuel.