Nothing will stand up to regular foot (or paw) traffic indefinitely. Even lawns get low track-like pathways through them when foot traffic is routine. If it is occasional or not a set route - difficult with dogs that patrol their territory - you will fare better. Beach strawberry (Fragaria) or Point Reyes ceanothus will tolerate quite a lot of traffic. And being native to much of the west coast, they will be quite drought tolerant once established. The strawberry is flatter, more ground-hugging - the ceanothus will develop a woody structure and can eventually get to be about 15-18" tall.

If the dog has a set routine through the garden (mine always did), you might want to consider formalizing their routes with something more durable, like wood chips or even pavers and use the groundcovers in areas the dog is less likely to romp through. Your choices will increase significantly.