Another way to do it, but it means an overnight each way, is through Calgary to Houston. We overnight in Houston one way and Calgary on the way back. When you think of the cost driving and storing the car in Seattle, it might be close to the cost of the hotels.
When we last went through Houston, most of the hotels were full, and when we finally found one, the shuttle busses had stopped running, so we had to take a $50 cab ride to the Ramada, so reserves at a hotel close to the airport would be a good idea.