Mid-August ... I'd plan two or three alternate routes because that's the middle of monsoon season when moisture and convection flow up from the south into the western states. Sometimes it's only diurnal -- flying early in the day avoids it -- and sometimes it's all day long. Sometimes the southern routes might be clobbered and northern routes ok, sometimes the reverse, or both, or neither. It comes and goes.
Even absent convection, flying early in the day is a good idea, as thermal turbulence over the deserts kicks in by noon.
The lowest-elevation route would be basically I-10 to Southern California then north along I-5 all the way to Seattle. That's also the longest route.
A 182 shouldn't have any trouble getting to 12,500'. At that altitude you can do I-80 through Wyoming to SLC, then Burley, Idaho - Boise - Baker City, Oregon - Pendleton - The Dalles - through the Columbia Gorge to the Portland area, then north along I-5 to Seattle. We did that in our 180-hp 172 from Kansas City to Portland in July a few years ago, no issues. Rock Springs WY is a good fuel stop. In the summer the density altitude can be 9,000' or more, but the runway is plenty long. If you're a Cabela's fan, stop at their home base at Sidney, Nebraska. They'll pick you up at the airport.
Or you could go up to Rapid City and see Mount Rushmore, then I-90 west through Billings, Butte and Missoula to Spokane, all the way to Seattle.
If you do come by way of Arizona, give a shout.