Wanting to build a coop and run that will fit over a raised garden?

Lots of problems here but some clarification might eliminate some.
You will have the world's most miserable hen with one hen one rooster.

 

Breeders often keep one rooster with one or two hens for the entire breeding season without problems. One huge key is that they use mature hens and roosters, not immature cockerels and pullets. It doesn’t always work as each chicken is an individual with its own personality, but it usually works out fine without a stressful hen as long as they are mature chickens. The only way you will know if it will or will not work with your specific chickens is to try it. I suggest you have a plan ”B” ready just in case.

I personally have not done that but many people have posted on here that they do. I hope all your raised beds are the same size. I don’t know how they are built, what the sides are made from or anything like that, but what you are talking about is a tractor. You might look through this section, it’s under the “coops” tab at the top of this page, to see if you can find something that you think will work.

https://www.backyardchickens.com/a/chicken-tractors-mobile-chicken-coop-designs

You are looking for something light enough that you can lift and move yet strong enough to stand up to wind and predators. You may need to build it in sections that you can move independently, say a coop section and a run section. I did that once with a tractor and bolted the two sections together after they were moved.

You may need a way to anchor it to the raised bed to keep the wind from blowing it off and to assure a predator can’t lift or knock it off. It’s hard to come up with a universal way to do that not knowing what your raised beds look like but if they are wood I’d probably look at somehow screwing them down. Maybe use strap hinges attached to your tractor that you screw into the raised bed when you set it and unscrew when you move it.

 

Sounds like a day job, maybe cover beds with a mesh enclosure to either confine or exclude
........but like PC said....lots of issues.

How big are beds?
What is layout of garden area?
I assume they will only be in there for the late fall to clean up and fertilize?