At first check the bystarter valve on the carb for proper function. And its wiring harness for breaks and chafe marks along its route.
It must be retracted when cold, and extend itself due to power flow of the running engine or just key on for a while. Test that at first.
There is lots of Information and Pictures and Videos out there about working and non working Bystarter Valves.
If it was extended cold, it would perfectly explain the symptoms. Dies when cold, runs fine when warm.
If the Bystarter Valve is working correctly, Carb. Cleaning would then be my next step.