There are several very good campsites along Butte Lake Road as you approach the national park boundary. Many of these are just off the road and experience a fair bit of dust from cars passing by. We found a spot along Butte Creek itself, just off Pole Spring Road (geocords: 40.586451,-121.291784). There is at least one other site just up the road (to the East) from this location as well. 

This site is beautiful - just off the main road, fairly flat, immediately adjacent to the creek, and very quiet + almost no traffic (have seen one car in two days). It easily accommodates our 32 foot travel trailer - you could easily fit several trailers in the space here. We also checked out the NPS campground but came back to this spot. The campground is dark and thoroughly infested with mosquitoes. The mosquitoes are also fairly thick this time of year (early July) at this site in the evening as well, but at least you're in an open space with sunlight!

One drawback to this area is the soil. The ground here is loose volcanic sand. I wouldn't recommend a heavy trailer rig or RV without 4x4 do anything but back back straight into this site. We bogged down a bit while turning around and needed to use our 4wd. 

No signal on our TMobile hotspot but the site has a fairly clear view to the North and our Starlink works excellently here (the NPS campground is heavily obstructed by thick tree coverage).