Love that avatar, Sara! Hubs and the "pups" thought it was hilarious, too. One concern we have though... surely that rabbit is not hunting wabbits?!? :shock:
I use wire nests also, and I line them with cardboard. I am in CA, so the temps aren't that extreme, but in the warm months I cut strips about 2"-3" tall for the sides, front, and rear- enough to keep the bedding and straw in as they build the nest. You will need to zip tie them to the wire or the doe will use them as building material! I put a cardboard bottom inside the nest itself, but if the heat is a real concern to you, you could put the cardboard underneath the box. Once the nest is formed and the kits are born you could remove it- but I think the cross ventilation from the sides should be adequate. You could even remove the side panels I guess.
Since you don't have hay or straw for bedding, you could substitute shredded paper. I like to use brown paper grocery bags that I run through our paper shredder, but white paper works too.