Craig, we've not been over that way but expect snow is minimal except in higher, shaded aspects. Burcham Flat runs from 395 north to Walker. The northern half is cut into the hillside. We've only camped near Lobdell Reservoir up the road to and past Lava Springs - lots of Basque aspen graffiti and bedrock mortars if you look hard. We've also camped a bit above the reservoir on the edge of a large open meadow. I think Lobdell is near 9000 feet. We camped up there mid May one dry year.
You may also consider exploring the Little Walker River Road. The elevation is a bit lower. After the ranchland, there are a couple roads to the right that drop down to campsites near the river. A bit farther is Obsidian Campground, small and primitive. Continuing on and over the bridge on the Little Walker, the road climbs and there are several primitive campsites. We've only camped up a spur road at the end - trailhead to Emma Lake. Before roads end, it opens up and there is a spur road to the left that crosses the sage to an aspen grove and campsite.
Get out and have a nice break.