I don't have a full answer to the question, but if you do feed the dog meat, one starting point would be to prefer meat that has less suffering associated with it. It is typically claimed that beef has less suffering per unit mass associated with it than pork and much less than chicken, simply because you get a lot more from one individual. The counterargument would be to claim that cows > pigs > chickens in intelligence/complexity to a great enough extent to outweigh this consideration.
I'm curious: are there specific reasons to believe that dogs need meat while humans (also omnivores) do not? A quick Google search finds lots of vegetarians happy to proclaim that dogs can be vegetarian too, but I haven't looked into details.
Here's a quick citation: http://pets.webmd.com/features/vegetarian-diet-dogs-cats
tldr: Dogs are opportunistic carnivores more than omnivores. They eat whatever they can get, and they'll probably survive without meat, but they'll be missing a bunch of things their bodies expect to have.
Previous Open Thread: http://lesswrong.com/r/discussion/lw/k9x/open_thread_may_26_june_1_2014/
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