I would drive South up the BRP from Asheville to the Pisgah Inn mp 408 and start there. That avoids a couple long tunnels on the ride up from town. From the Pisgah Inn you can ride out as far as you care to, go to the high point at Richland Balsam mp 431.4 which is 23 miles from the Inn. Turn around ride back. There is a good water spring coming out of a pipe about a third of a mile beyond the highpoint on the right. There are no services on the BRP once you leave the Pisgah Inn.
Also, you can either start at the Folk Art Center mp 282, East of Asheville on the BRP and ride up to Craggy Gardens Visitor Center (water, restroom) and back. It's a popular ride but can get some traffic. Alternately, drive up a ways on the BRP, then ride up to Mt. Mitchell (the highest peak east of the Rockies, water and RR) and back.
Hope that helps. Best to ride earlier than later due to afternoon T-storms.