I would be thinking its a carb or intake issue something fuel air related. A cold engine requires more gas, so if its not getting enough air somewhere it might affect it more when its cold? A bad air filter, maybe an air bleed in the carb? Did you check the choke to be sure its closed and opens properly?
It could be fuel to when I first bought my old 79 wagon it would run for a couple miles then stall if I let it sit I could start it and drive it a little further before it did it again. We changed thefuel filter the carb the fuel pump the ign module, I finally had someone check for gas at the carb by pulling off the hose and sticking it in a jar right after it stopped. Just a tiny trickle. It was the sock in the tank.
Somewhere I read about blowing air back through the gas line to listen for bubbles in the tank. I used an old bicycle pump with t he fitting cut off. After I did that it would run ok for a day or two. So then I had someone pull the hose.
Some gas just leaks through the clooged up filter I guess but only enough to go a few miles.