I don't know the laws and don't advocate breaking any... But whether out in the middle of public land, or in my own back yard, I'm not going to ask permission to bury my pet. Just make the hole deep and or covered in good sized boulders so scavengers or busy bodies won't be able to find it or mess with it.
I worked in a vet hospital when in college. I held dogs as they were being put down. For a calm and trusting dog, it is very quick and peaceful. But other dogs, like mine, just hate everything about the vet. They shake, pee, and fight it the whole way. No way my dog was going out like that...