I mow in between the boxes. ANd my kids cut the grass that my mower cant touch using scissors. I use a chicken wire covering until the plants reach the top of the wire and or get crowded in.

Maybe exercise pens (like a portable fence, used by dog people at indoor events and shows) would work for you? An on-line search for "ex pen" or "exercise pen" and "puppy" or "puppies" or "dog" or "dogs" should return lots of options. They may seem expensive (depends on how tall you need them to be), but I'm still using the very first one I purchased ten or so years ago and it's going strong.

Sometimes people use X pens to keep puppies in a "potty-proofed" room of the house, like a kitchen, and to keep the pups off of hardwood floors.

I use one of my X pens outside as a clothesline, even, by using clothespins to attach the clothes to the top of the X pen.

Cynthia H.
in el cerrito