Somewhere on this board is a thread that addresses that. I think something was suggested or applied for to the FAA, to be able to use a specific call sign, like we do with Angel Flight when we have a patient aboard. It was denied, for various reasons, mainly not being necessary.
I do put "Animal Rescue Flight for PilotsNPaws" in the Remarks section of my IFR flight plan. So far not one controller has metioned it, although I have heard other pilots talk of being asked about it by ATC
Here's that thread: viewtopic.php?f=9&t=300