Since it's gasoline, it changes the normal diag of a propane fueled engine. Sounds exactly like any old chevy from the 80's with carburetor. Guy would start the car cold, then go for cigarettes around the block. The engine started fine cold, and had enough heat for a restart WITHOUT the choke. But when he got back in the car a few minutes later, the choke came back on full and flooded the engine....if he touched the accel even slightly while cranking. The only way you could do a warm restart was to floor the gas pedal. The car started but smoked and ran rough til it cleared. But you never had an issue if the car reached operating temp.
Youre choke system may have lost power to it's internal heater. Check that first. Make sure the linkage is opening the choke when the gas pedal is fully depressed. Usually the choke cant close unless you open the throttle a bit but some carbs did not have that feature. You may be running rich or worse have a buildup of carbon in the manifold. Last one I did had an inch of carbon under the carb!
If you dont want to spend much more time, just make sure the operator doesnt touch the gas on warm restart and floor the pedal if it doesnt start right away.