I think I've heard that it says it's going to cost 500 CSC for your free change but it doesn't actually take it. Not that I would be willing to test that on a pet out of the pound since it doesn't seem like it's supposed to work that way.

I've thought of doing that as well, just cashing my CSC out in cards before trying so even if it would cost the CSC I wouldn't have any to take.

It would make sense because the original owner was the one who had their pets name taken so you'd think it would be the users perk and not just the pets?