I would rather not going with an enclosed trailer. It's PITA enough to strap down the Z06 on an open, I can't imagine an enclosed one. Plus, an enclosed trailer is whole lot heavier (and much more expensive); I have a plan to downsize the tow vehicle which doesn't go well with an even heavier trailer.
I don't want to store a car inside or on a trailer; a stolen trailer means I will be losing two vehicles in one swipe. Not to mention that being an engineer, I can't see the trailer suspension and tires being loaded all the time. If I want to put another car into the garage, I will get another 4-post but not storing a car in or on a trailer.