I haven't ever had those modules act like that. They have always caused it to die when running and then wouldn't restart. After 15 minutes it would start normally.
I go with fuel. I know it is winter, but is it warm enough anywhere around the engine to cause vapor lock? They say this newer fuel does it easier. Does it have a mechanical pump? Wondering if the pump bleeds off after that long and has trouble pulling fuel when the pump body is hot. I think fuel is the direction to go.