I live in New York, not Seattle so I don’t know what it’s like there. But here’s what we do in New York. My husband flags down the cab while I stand about ten feet away with the dog. When the cab pulls up and realizes there’s a dog, they often just accept it and let us in. I’ve only had one driver say no.
I read about that trick in MetroDog, the book about living with a dog in urban environments. The author also suggests standing by a parked car so the dog is hidden behind it when the cab driver sees you.
When I’m alone I often put the dog in a bag. He’s small enough to fit hidden in a bag. Some drivers never notice it’s a dog bag. He’s also small enough to legally ride like that on the Subway. Lucky us.
In New York we also have special pet taxis you can call. I haven’t tried that though.