I'd sure stick with the .257 of the two you mention, and the use you intend. I'm not shot with the 6mm bore size, though many are, for big game. If you insist on the 6mm 06, you might look at the .240 Weatherby as it's about the same thing.
Since you are cooking this up from scratch, I'd consider moving up to the .270 Weatherby. Same case as the .257, lots more punch with heavier bullets, easy to shoot, nearly as flat as .257.
