When I looked at getting the CU to heal me for its gains, it healed me far too infrequently to make for good gains.
Fastest gains I get on dogs is like this. For the start, I do an all stay and then discord the dog and repeatedly cast poison on it, making it cure itself, this is assuming the dog's healing and anatomy are mid-level already, say 75 or so. When it gains to where the self-cure is succeeding first-time virtually always and the gains have slowed down to nearly nothing, I switch over to having a rotting corpse poison it (this is in the neighborhood of around 90 skill). If I had a nox mage or a means to do the higher level poisoning myself, it wouldn't be necessary to use the rotting corpse.
On the rotting corpse, at first it's a little troublesome because the dog is still a bit low to have a great success rate on that poison. If the dog is taking too much damage, it can help to discord the rotting corpse in addition to the dog, and remember that the rotting corpse trains up as it hits the dog so, as time goes on the situation can shift. Training the dog this way, going for max speed gains, requires second by second participation on your part because you have to magery-heal the dog when it is not poisoned, and try to force it to do its own cures, while not having the dog die from the damage during the process. The benefit of magery healing it is that you won't accidently cure the poison, but you do need to have Vet heals available because you will occasionally need it to save the dog's life. You sometimes have to cure the dog and heal it up fast if its self-cures have failed repeatedly.
Some people leave the dog alone in the room with the rotting corpse, allowing the dog to kill them as they respawn, while they themselves go do something else. This does in fact work but it's not as fast overall, and leaving the pet alone carries it's own issues (loyalty lowers over time and has to be refreshed, and someone else may show up and start killing the rotten corpse for their own reasons).