FYI,
Bartlien will manufacture a barrel for you with a shank diameter larger than 1.25", at no additional cost. Call them up and ask them if they will keep the breech shank "as large as possible". Large as possible means the diameter that the breech first cleans up to run true in their lathes. When I ordered my barrel, they told me to expect a breech shank diameter around 1.290" to 1.300".
The blanks they use to manufacture their barrels with evidently come with diameters a little larger than 1.300". When the exterior is turned down to running true to center in their lathe, they typically measure 1.290" to 1.300" in diameter.
Oh..., this was for a stainless steel barrel. I can't speak to their CM barrel blanks. Best to talk with them to confirm what is possible.
My .338 SS Bartlein barrel has a breech shank diameter of ~1.303". I wanted a little larger than 1.25", which is the minimum recommended by BAT. Some gunsmiths feel the additional breech shank diameter is helpful to stable accuracy...