Cladosporium paeoniae tends to occur more on leaves/stem I think. My guess would be Phytophthora cactorum which is generally fatal to the plant and usual recommendation is that infected plants should be dug up and destroyed, and the soil replaced before replanting. Personally I would cut away any black/soft parts of the roots, soak the rest in a general fungicide then pot any healthy pieces up in a very gritty compost. 'Coral Charm' in my old garden re-grew from fragments of root left behind when I moved a plant so you may find all is not lost. (But keep any resulting plants away from your sprinkler-wielding friend; peonies are more likely to get blights etc if the foliage is wet)
