Does cargo heat failure require a diversion? What about if there are live animals in cargo?
Does cargo heat failure require diversion? It depends on the cargo being carried.
For your question about pets, then yes.
Council Regulation (EC) No 1/2005 regulates the protection of animals during transport in the EU, which itself is in accordance to IATA Live Animals Regulations. The guidelines based on that EU regulation say:

Commanders, and loadmasters and aircrew under their authority must ensure that the aircraft and equipment are suitable for carriage of the animals concerned in the manner intended. They must ensure that the animals are loaded, carried and unloaded in a way which will protect their welfare.
In particular they must ensure that the animals are accommodated in accordance with the IATA LAR, and that an appropriate environment of air quality and quantity, temperature and pressure is maintained whilst the animals are on the aircraft.

The ability to ventilate and regulate the temperature of an aircraft belly hold varies considerably. The arrangements vary between individual aircraft depending on the make and type, the compartment, and whether or not ‘standard’ or ‘special’ specifications have been fitted.
It is therefore essential that the aircraft operator, who should know the individual features of each of its aircraft, is satisfied that the belly hold to be used can be sufficiently ventilated and maintained at a satisfactory temperature during all stages of the proposed flight. This assessment should take into account the species, number, weight and size, and volume of the animals to be carried, and the heat and moisture that they will produce. Other cargo being carried at the same time will also need to be taken into account.

Even the Minimum Equipment List of aircraft takes that into consideration. For example, the aircraft can be dispatched with an inoperative cargo heater or ventilation fan, but not if there's an animal in that respective cargo hold.
All that confirms the pilot's action in the story you mention.