I read the document from PALS. If you want to read it, its easy to find under 'documents' or something via the PALS site.
I couldn't correlate it for use with animals... at first glance. seeing how my furry passenger flight is not time sensitive, I'd feel weird in my cherokee at 70kts claiming 'compassion'. If I had a hurt dog on a clock I'd simply do it. Time would then be important. Organs, blood, medical tools and supplies can use 'compassion' and they are time sensitive.
keep in mind that the FAA document about it suggests that since its new, not all ATC is aware of it and may reject it. We as pilots must go along with that, even if we are compassion with a human-related flight of PALS.
so no, I won't use it.