It sounds like air getting into your fuel line.
I had a problem that sounds similar. Engine starts (similar environmental conditions as you), runs for minute or two, then stops. Crank and crank and crank, no good. I ran the fuel pump
opened the bleeder valve to purge air (Kubota block has a small rotating valve that I open to purge air, really slick), then starts. Sometime stops again, sometimes runs for a while.
Turns out it was an air issue I caused. I removed the generator
and filter months before to rebuild
and didn't do a good job sealing off the fuel line from tank to generator
(FYI: Threading a 5/16" bolt into the end of 5/16" fuel line will not seal it very well!). In addition, the generator fuel input line was T'd off the main engine pickup at the tank. Air was being sucked through the end of the generator line, past the T and into the engine.
I fixed it by removing the T and giving the generator its own pickup tube in the main tank (actually, I designed and made a new tank so each device, gen, engine, heater
has its own pickup and return.
I never was able to confirm that it was air because I never saw air bubbles (or was not smart enough to look). All I know is that when I replaced the tank and hoses, I no longer had the issue, except that one time when I didn't tighten a hose clamp enough and air was getting sucked in, causing some run failures. Double check hose connections to make sure they are tight enough.