Well, there may be a bit of heat soak going on as well….Winter fuels used in warm/hot weather tend to be more volatile, and may even be vaporizing in the bowl, which might explain why it seems to run better if you pump up the fuel bulb. Again, seems like this engine runs OK, so I wouldn’t go digging for ghosts that most likely are not there. There would be running issues if there was an air leak/fuel pump problem most likely…Seems like the engine runs fine once restarted and past its initial rough running condition.
I guess I would just pump up the fuel bulb before I restarted when hot/warm to alleviate this condition.