When I first read, the first post, my thought was the start solenoid or relay in that circuit. I see that that has been discussed, but I do think a contact somewhere would be the issue, or as
@Darren and others have eluded to, something electronic. I would suggest applying heat (hair dryer, heat gun but gently) to one component at a time,ie, the starter/solenoid, ECM, etc. The ignition push button starts it all and any contact anywhere can be affected by the ambient temperature, and see if it goes from no go to go. I had a plane once that was intermittent, and it was the ignition switch itself. If the load test on the battery shows good, and those connections are good, I would move on. You could try an interior heater since the pushbutton and some electronics are in the vehicle. If this does nothing, look further under the hood. Even hearing a relay clicking is, although a good sign, not proof it is switching.
I am assuming, although I could not find it mentioned directly, that the car is not even cranking, yes? It goes from cranking normally and starting when warmish, then won't even crank when cold? If I misunderstood, then disregard.....
A great virtual support crew here! You are in good hands.
Just my $0.02