A picky question, but what color is the cover? I think black on a hot day might be a bit too much radiant heat from the top, especially if it's well-sealed from wind. Lighter would be better.
I'd be interested in how much air circulates inside the bed. I wonder if maybe a video recorder secured inside with some loose, dusty hay and a test drive at highway speeds might offer some insight there. Can the cover be undone near the cab, like one or two snaps to let a little air in?
If the cages are secured with a load bar, going without the tailgate or replacing the gate with a Pro-Net type closure might increase air circulation while keeping the rain off of them.
Big ice bottles, secured so they don't roll around the cage would be a must. I would try a shorter test drive somewhere in the heat with one or two rabbits. You'll know in 30 minutes or so if the heat, wind, motion, etc are stressing the rabbit. Keep an outdoor thermometer back there and check it after you drive somewhere to know how hot it is at highway speeds.
