You probably have to get out past the Patton Museum at Chiriaco Summit to find suitable shootable BLM land. Although I have seen folks camping in RVs on the dirt roads just off the road in from the south side just out the south entrance, there was nothing in the way of suitable backdrops until you were well inside the park boundaries. (I'ld also bet that was private property too)
My impression is that most if not all of the land relatively close in on the north and west sides is private property, and not reachable by passenger car type vehicles, plus off to the northwest you probably get into San Bernardino County no shooting areas.
So that leaves off to the east side. Take plenty of water, as even most of the Park campgrounds are dry. Also a shovel and some extra matting to help get out of sand traps.
If you haven't seen it the Patton Museum has a very impressive 3d map that the Metropolitan Water District put together when they surveyed most of that area for the water projects. They probably also have information on the roads left from the War and water projects.