I stapled the chicken wire to the wood base on mine and it worked fine. Hardware cloth isn't much larger in diameter, a lot stronger too. You should be fine along the bottom.
I moved the nest boxes down on my old coop by turning the panel over they were attached too. Looks like the back of yours was one piece of plywood cut to match the arc of the hoops. Might could cut out the middle section leaving the tapered edges and turn it over along with the nest boxes. Then add a couple of 2x4s vertically bridging the seams to put it back together. If they would be too low, cut the desired width off the top of the center section and attach it below the nest boxes. It would save having to buy a new piece of plywood and would be already painted. Might need a couple of 2x4s depending on how it's framed in the back.
I moved the nest boxes down on my old coop by turning the panel over they were attached too. Looks like the back of yours was one piece of plywood cut to match the arc of the hoops. Might could cut out the middle section leaving the tapered edges and turn it over along with the nest boxes. Then add a couple of 2x4s vertically bridging the seams to put it back together. If they would be too low, cut the desired width off the top of the center section and attach it below the nest boxes. It would save having to buy a new piece of plywood and would be already painted. Might need a couple of 2x4s depending on how it's framed in the back.