If I was doing that I would go as far as Bulford, ( just before Amesbury ), then turn right ( West ), and head past Larkhill camp. It's a little over 4 miles to a right turn onto the B3086 which brings you direct to the visitor centre.
The bigger problem would be getting back to your route. As far as I can see you can either reverse the above track, or from the vistor centre take the A360 southbound and cut across to the Woodfords. That does, however, mean negotiating the Longbarrow roundabout as the A360 crosses the A303.