Almost every trip is worth doing once in your lifetime. Living near the BRP, and not far from its junction with the Skyline Drive, I've been on both often over the years. A 25 foot C with no toad should have no problems. Do watch for clearance issues with tunnels, however.
The Wally's in Roanoke is not all that close to the Parkway, however. I'd highly recommend springing for the $16 overnight fee at one of the no-hookup campgrounds right along the parkway instead.
As others have pointed out, both the BRP and the SD are slow drives. Both would be closed entirely for snow, but this is not likely in April. Stop often and enjoy the views.