Beechnut,
Yes that's correct, there's no bleed air (except for the engine anti-ice). It's all CAC's (Cabin Air Compressors). I presume the dog was in the FWD cargo as that is the Cargo compartment that you have manual temp control but if the FCAC (FWD Cargo Air Conditioning) fails then you'd lose the ability to control the temp.
Could have been in the bulk cargo as well but the temperature there is automatically adjusted to maintain 19c-21c.
Good call on the crew to divert. Nice to see that life still matters more than profit in some parts of the world
