I GM'd healing on one of Cu's a little while back just to see if I could do it. The Cu has to heal itself in order to raise it's own healing skill, if you heal it then your vet/lore skills are being used. The Cu's healing skill will raise through normal game play until it gets to 90. After this you have two options: 1) In order to raise the skill above 90 the Cu will have to heal itself of Deadly (lvl 5) poison which can normally happen when fighting Rotting Corpses. 2) If you have the bard skill Discordance you can discord your Cu and let the pet heal itself through normal fighting. Discordance lowers all of the skills and stats which lets the healing skill continue to be raised through normal fighting/healing but the Cu will not be able to fight as high a level of creature as it would when it's not disco'd.
I went with option 2 because I have a bard/tamer. I fought Tsuki Wolves and harvested leather and seeds from them while training healing on my Cu. I can't remember how long it took me but it wasn't more than a week of a couple of hours a day of playing.
In all honesty, it's something I did just so I could say I did it and to have at least one Fully GM Cu but it's not something I will ever do again because I never really saw a huge difference during fights as I don't rely on having my Cu heal itself. I do the primary healing with vet or greater heal and the Cu's healing skill for is secondary (nice to have but not necessary).
)seems to go up a bit in there, sometimes i have to run away and let the cu pick up aggro, so it has to heal itself for a bit, but bring a friend or 2, sometimes the cu is too good at grabbing aggro