I think Bull Run & Bull Road are interchangeable. It's marked in the PNF KMZ file as Bullroad (all one word). It's the 202 trail.
Actually, there is a way to do it where you'd skip a lot of the northern section of trail and it wouldn't look so rocky. If you came down Bradshaw and then hooked over on Tuscumbia, that cuts off a lot of the northern part of the 202. You'd end up in the same spot if you went all the way down Bradshaw and then took Battle Flat over.