Most cat litter is made from clay (e.g., bentonite) that acts as an absorbent. Since clay typically contains elevated levels of naturally-occurring radionuclides, large amounts of cat litter can be measurably radioactive. Shipments of cat litter have been known to trip radiation monitors.
I have been reading a lot about mycelium helping plants lately. I was thinking of growing King Stropharia (I believe the same as what you called burgundy mushroom) in the shadier parts of my garden; I can use hay to get them started. My biggest problem with growing mushrooms is that the vast majority of them want hardwood- as chips or for plugs- and we simply don't have them in our area; it's all pine and juniper. There is a variety of oyster mushroom that grows on softwood, which I may try. We also have morels, but they seem to be very difficult to domesticate.