All the covers I have purchased to date warn about transporting the motorcycle with the cover on. The idea here is to keep the flapping/vibrating cover from damaging the finish of the motorcycle. A "dust" cover will tear in a relatively short time, an "all weather" cover, being much heavier, will do more damage as it flaps/vibrates, but may be viable if you are willing to spend the extra time to chinch it down and place soft towels on the painted parts first. Some covers have grommets on the underside that can be used to run belly ropes to cut down on the possibility that the cover will flap/vibrate; any place the cover is a loose fit it will flap/vibrate. Any way you cut it's a bit of a dice roll. If you have the cheese to have a custom cover made, call the nearest marina, they always have a service (if they don't provide it themselves) that can whip one up; "top" shops, may be more local and should be able to do the deed as well.
An enclosed trailer is the best protection, you can also sleep in it. If you are committed to an open trailer I recommend that you cover the motorcycle when parked and let it enjoy the ride, uncovered, when moving; it gets to see what you see.