Houston to Seattle is 2350 miles by fastest route. This is definitely a 5 day trip, driving about 10 hours a day including fuel, food, and rest stops. This route involves a considerable distance on non-Interstate highways.

I-45 to Dallas/Fort Worth, US-287 to Amarillo, 287/87/385/64 through Dumas and Hartley to Raton, then I-25/I-80/I-84/I-82/I-90.

Taking Interstates all the way will add over 300 miles to the trip.

One potential problem is finding pet-friendly hotels that will allow 2 large dogs. I would plot out the route and start calling hotels along the route to determine their policies. I show the suggested overnights as follows:

1. Somewhere between Wichita Falls and Amarillo TX
2. Pueblo CO
3. Green River WY
4. Boise ID

5 days should not be a problem if all you want to do is get from point A to point B. However, if you wanted to make any significant sightseeing stops and/or detours, you need more time.