Ski Shores Waterfront Cafe on Lake Austin is dog friendly as long as they remain on the leash. Amazing burgers and good fries. Fried okra is yummy too and the pizzas aren't bad. I think there's something called the Border Burger that my whole family loves. Ski shores has a small covered patio but it is right on the lake so you might avoid if it's really stormy and I think they probably even shut down in heavy rain since the majority of seating is outside. It's a tiny place and usually crowded in the summer so be prepared to wait a little for your order if you go then. Right now they are most likely in off-season hours which are Thurs-Sunday.
These venues have al fresco dining or bar menu service:
Don't know about the dogs, but I imagine this would be difficult at the better places.
Mansion on Judges Hill
South Congress Cafe
Shoal Creek Saloon
Dog and Duck