Hi John. Long time since we've heard from you. Hope you're doing well.

Based on everything you've said, the PNP neutral safety switch would be the top of my suspect list as long as you're happy it's not the starter motor/solenoid (like tapping hard on the solenoid with a 2x2 doesn't help?). I've had a number of nuisance, intermittent no-starts from that switch over the years, just not the hot and cold issues as you've experienced.

Locate the open pink wire remote start connector in the left fender wiring harness and jumper that to the starter solenoid (blue-green) to bypass the PNP switch. If it will now crank in KP-III then you've pretty much got it narrowed down to the switch (or perhaps the wiring to it).

The PNP switches are a real pain to get to, having to raise up the shifter assembly in its housing, and it's easy to break the two thin plastic mounting studs if you're not extra careful. Bending the star lock washers open one tab at a time using a pointy awl works for me. Once the switch is off and to the bench, you can pry it open for access to the contact area to clean it up and re-lube it with something like dielectric grease.

Be prepared for a solder connection to break in the process -easily fixed. I've never had to replace a PNP switch, always been able to resurrect them with a cleaning.

Only tip I've got for you is to use acorn nuts on the plastic studs for re-installation rather than the original star lock washers -so much easier to remove if you ever need to again. I forget if it's SAE 5/32", 1/8" or 3/16" acorn nuts that work fine as long as you only do them finger tight so as not to strip the plastic. Using a socket on a handle helps keeps them straight for threading on.
--
Dave -still with 940's, prev 740/240/140/120 You'd think I'd have learned by now