If you're up for a slightly longer drive, you might want to check out rivers in Idaho. Permits for the Snake through Hells Canyon aren't too hard to get, and cancellation permits for the Main Salmon aren't too hard to get either, assuming you don't draw and are flexible with dates.
Not only that, there are a lot of sections of the Salmon that can be combined and run like a multi-day trip, no permit, no fees required. The North Fork to Middle Fork (confluence) section of the Main Salmon is an great 39 mile run, especially from Pine Creek to Panther Creek (similar to Brown's IMHO, similar to Browns depending on the water level albeit with a road running the length). This stretch is roadside, but you'll see very little traffic actually on the river - other than a few locals, the traffic is all coming off the Middle or putting on the Main, except on a few summer weekend.