I have had this issue with 7 series Toyotas with the asain fuel system. You are correct that the fuel shut off solnoid does have a tendancy to stick when hot. Unfortunenately the only cure seems to be replacing the entire regulator. I don't know why, but testing & even replacing just the shut off solnoid never seems to put the symptom to bed.

To test the solnoid, have the engine running & get it hot. Turn off the propane bottle & let it run out of fuel. With the bottle still off, crank over engine. While cranking,turn bottle on. If it's the same problem I've run into, the engine will fire right up. It seems to be the back pressure from the fuel that stops the solnoid from working. I've tested the solnoids (they've tested good) & even replaced them, but the sypmtoms still remain.

Hope this helps