I think (subject to check) that Florida requires the plate to be attached to the vehicle and be illuminated with a dedicated light for night operation. That would preclude relocating the plate to the bike rack if Ohio has a similar law.
I've hauled bikes all over the country on a trailer-hitch mounted carrier without an issue. I think cops usually give cyclists a break because we're generally upstanding folks. If there is another violation involved in the stop, like speeding, the officer may well add on a charge for obstructing license plate visibility for good measure.
What about a rack that mounts on top of the car? All you have to do then is take the front wheels off...
My left shoulder is frozen. A roof rack is out of the question.
Some people are like a Slinky ... not really good for anything, but you still can't help but smile when you shove them down the stairs.