I wouldn't suggest goat milk or greek yoghurt for ramping up good gut bacteria. I would assume both are pasteurized, not to mention milk sugars involved.
I would recommend choosing a high quality human grade probiotic in capsule form, given to your dog on an empty stomach for maximum benefit.
You could give your dog a capsule once a day for a couple of weeks then taper off to a few times a week thereafter.
There is way more benefit in one capsule than your dog could ever derive from any milk or pasteurized yoghurt product.