Assuming your car can be towed on a tow dolly (check your owners manual, most front wheel drive cars are fine, but AWD cars a big no), full trailer will be a little easier to get around with, primarily because with a tow dolly you can not back up, at all. However, I'm not sure that's something that's worth a $300 difference. What I would recommend, and what I've done in the past, is to talk with the people at the actual rental location and see if they can reduce the cost. Since you are already spending a good amount of money on a cross country rental, they may be willing to work with you and get the price for the full trailer closer to the dolly price.

You can see Thunderstorms at any time of year with the right conditions, its not something that's a purely seasonal phenomenon. However, it would be past peak season for severe weather, and even if it was the peak, the odds of seeing a twister on any given trip is quite low.