Have you considered Rainier National Park? It is BEAUTIFUL! beyond compare. John Muir thought it was the most majestic place in all of the US near the end of his life, if I recall right. He spent a lot of time here. I have to agree that Yellowstone looks a lot more like a tourist destination unless you plan on going damn deep into it.
Anyway, you have all that, and literally just across the road (turn left to go to Rainier, turn right and you are in national forest) you can go shoot all you desire. Just find a safe place to do it; there are LOTS of places out there to go shooting on public land.
Rainier has its fair share of tourists, but mostly they go to the pavilion and visitor center. They hike the meadow near the top. If you hike the thousands of other trails there, you may not run into anyone at all, and one is over 100miles long.
Just a suggestion, I live close to this volcano, and I've seen my mom's old pics of Yellowstone and it was crowded as hell back in the 70's. I hear lines to get in can be daunting as well, and I've heard it just isn't what it used to be. Also, with these national parks, the more people that visit, the more draconian the laws and rules governing what you can and can't do there are implemented. Most good natured, some goes too far though IMO.
Near Rainier, there are also several ranges, public, private and military that are available to civilians too. 3 are within a 10mi. radius of me with 2 about 2-3 mi. away. There is a 1000y range out there somewhere too, near the mountain I think, or at least on the way? Cross the mountains and you have desert. Find public land there and have at it.
The one destination that nobody considers is Teddy Roosevelt National Park in the Dakotas... Beautiful, I mean gorgeous, and it has these awesome canyons and wild buffalo. I'm POSITIVE there are plenty of great places to shoot off the park land, and that place is worth checking out as nobody really knows about it and it is about as pristine as it gets. I never heard of it until I drove through it. Wanna go back.