I really see no reason that a transformer can not be laid on it's side. It might have to do with the position of the internal transfer switch, and having it upright is a predictable way for the transfer switch to open and close properly. Or it might have more to do with cooling air currents?
I can do what I want with my transformer, it is just a transformer, not any electronics in it to change the voltage or anything. It does create heat. You do not want to store towels on top of it, as that will keep the heat inside it, and it does not like that!
Have fun camping!
Fred.