I checked on Spooner Lake but there is no campground, apparently that was a rumor. The Mt Rose campground is closest, about 15 minutes from Incline Vlg and has some trailer sites, up to 35'. $13 and takes reservations. Even tho it is at 9,000', we have pulled our Casita
over Mt Rose with no problem. Nevada Beach is at lake level, but farther away and more expensive, $22-24, takes reservations.
In Washoe Valley; Davis Creek campground is in Pine forest, costs about $14 but does not take reservations. Washoe Lake campground is in the open, sagebrush by a small desert lake, $12 and takes reservations. These are 35-40 miles from Incline.
I live in the area, so PM or e-mail me if you want more info.