Road shoes should be fine. Keep in mind that running/walking *down* from the top of the incline is by far the hardest part of the trail, traction-wise. For the Ascent, you don't need to worry about downhill, and climbing through "the W's" you most likely won't be running much.
Although the Barr Trail is fairly rocky, its not extreme. As long as you keep your eyes open, there isn't really any challenge finding a reliable, safe place for each footstep.
The only need I can see for trail shoes is if you want stiffer soles for more protection against rocks. But the only place where the trail gets seriously rocky is towards the end, and (compared to other mountain trails/roads) it isn't that harsh.