How about chucking the dowels into a drill chuck, any 3 jaw chuck should work, then setting 2 indexing pins that would drop into the gap between the jaws, 1 @ 0* and 1 @ 60*. You set it up so that as you spin the chuck one of the pins will always fall into the gap and lock it in place. That will allow for your even rotation, the chuck is never under power, just used to hold one end of the dowel and to locate your stops. The other end could be supported by drilling a hole in a block of wood and sliding the dowel into that hole. Then you could use a sliding brace to support one side of the length of dowel while you run a straight bit down the other side with your router. As you turn the dowel and hit the flat spot you move your sliding brace into position to contact the dowel again. Set your router on skis and let that guide your router for a straight run, or build a track that doesn't allow the base shoe any side to side movement, only lengthwise movement.
I am interested to hear what the rest of you think of this idea, it sounds straight forward to me but if you think its stupid, please say so, but also please explain why.