I have heard of DLM but that only works in the large coops like yours I think. I have 3 smaller coops (2 4x4 and 1 4x10). They are chicken sized and not walk in size. So, I think they need to be cleaned out every week, and spot cleaned more often than that. I'm definitely thinking of going to one large coop like yours. Meantime, I'm pretty happy to hear I can use hay. There would be some great benefits to it.
Hay gets moldy pretty quickly; I don't believe it is safe.