I don't think insects or deer really harm TPs at all, but there are fungal diseases which younger plants are very vulnerable to if the area is not kept dry and sanitary. I think the disease is called botrytis which is similar to dampening off in young seedlings.
I've never tried rooting cuttings before however I do have an idea of building a raised bed box over the entire TP so that soil can be backfilled onto it, and then the buds on the TP can become shoots which also send roots down as well, and once there are many roots formed, each of these shoots can become a new division in the fall once they are hardened off. This is basically a form of air layering, but with the entire plant, and I think it's better to cut off all the fower buds as they require a lot of energy to produce. I think cuttings don't always have that great of a success rate starting with no roots, and they may take a lot longer to become established plants. I just like them being well rooted before creating the separation which should happen perhaps mid September.
Here's a forum thread of someone who took cuttings successfully so I do believe it works.
Tree peony cutting, new root produced - Peonies Forum - GardenWeb