I've had my eye on this one for a while and believed I could do it in 4 hours, based on many hours spent on these trails/routes in Taos Ski Valley. Went solo from Bull of the Woods trailhead with a minimal kit (the weather window looked good). Filled up water in the creek that becomes the Middle Fork of the Red River (which you cross just before you begin the "13 switchbacks" up to the ridge that takes you to Mt Walter). Basically just hit lap atop Wheeler and kept going. The traverse went well--I was able to complete it about 12 minutes fast than I had expected. Tyler King is right that the headwall down climb from Rich Peak is at least bordering on class 4 (albeit only about 10-15 meters of it). This traverse is such an epic route; more people should do it.
Bull of the Woods TH to Wheeler summit (via Walter, of course): 1:44:52
Wheeler to Kachina: 1:33:57
Kachina back to Bull of the Woods TH (via Rubezahl-Twining Rd): 32:25