Pisgah is good because they have a large outdoor area. Been awhile since I've been to Wedge, but they had a fairly small tasting room/ but large area outside that you can keep the dogs (but they have a great IPA and some good seasonal beers). Wicked Weed, downtown, has a large outdoor patio in front and is rated as dog friendly. Don't know how adventurous you are with beer, but Wicked Weed also has a second bar specializing in sour beers (Funkatorium). It doesn't have much patio space though. Asheville Pizza and Brewing in downtown has a pretty large patio (they have good beers and pizza). If at times you've had too much beer and need to drown out with a protein high, 12 Bones is a good BBQ place that's also rated highly for dog friendly. And if you can part with your dogs, I would suggest you have a dinner at Lexington Ave Brewery: they are pretty trendy and have an OK range of beers with food/entertainment.
It's funny how many breweries keep springing up in Asheville. Asheville is my hometown and it just had Highland Brewing before I left for college. Once I got into home brewing was also about the time Asheville started being considered "Beer City" 