Exactly how would this work?
There are 18 dog breeds. I don't know if there's a limit to save files but I see 6 slots. You can buy extra dogs in save files, but let's see... 1 dog costs 1 coin. I don't remember how to earn coins, but in my save files with all races completed (one with all platinums, another with 3 golds) I had earned only 7 coins. You must actually play the game to get coins to buy dogs. In a single save file you can only have up to 8 dogs, including the dog you started with. To get dogs of 18 breeds you must start at least 3 new games (1st with 8, 2nd with 8 and 3rd with 2)... The thought or actually running this scares me.
The petting itself takes little time, sure. But to pet dogs you need to buy them with coins first. To get coins you need to play the game, and that's the time-consuming part.
Just checked the adoption menu, there's no option to remove dogs you already have, so you can't cheese it like that.
This doesn't seem possible, the game only offers 6 save slots. I don't even know where they're kept in the game's folders so I can't do a backup.