Faye, you are right, you'll either have to fence the chickens in, or out. How big are your beds? You can do the high fence, or you can do a tunnel over the top, like an inverted U. There is a type of PVC that is quite flexible, comes in 20' lengths, and you may be able to bend it into ribs that would fit the contours of your beds. Then you could cover it with standard fencing, or even use bird netting, and lift the netting off when you access your beds. You'd have to be sure it's secured well enough that your girls can't get tangled in it. You may find that a 4 - 5' tall fence will be enough to deter them, as they have the rest of your yard to play in.
I have some lavender plants in an established bed, and they leave them alone. Perhaps you could plant them, then cover with a chicken wire cone for the first season. I also use a wire border fence that works well for areas of spot protection. It has wire that sticks down about 8" to push into the soil, and the wire border has rectangles that are about 3" x 6", and is not much more than a foot tall, but they don't try to get inside it. I'm sure if it was just edging a bed, they'd hop right over, but it works great for protecting a single plant.
Pullum: Can you do a deep litter mulch in your run? Some folks use the refuse from tree trimming companies. Other options might be fall leaves, grass clippings, stable litter, hay, straw... any thing that is high carbon, or a mix of high carbon and fresh green (grass clippings) would provide a great substrate for your run.