Kinda like Pokemon Go (only not).. It would have the same UI as in game, except you would just have access to Pet stuff ONLY.
Would you pay an extra, say, 5 bucks a month for mobile access to help pay for the cost of that extra server usage? If not, do you think it should even go fully mobile?
IMO I think it would be a great idea. As someone who has capped his 1500 pets (900+unique), I would go for it.