Most states don't allow shotgun shot size larger #4.
While it is legal to shoot a turkey with a rifle I ask why? You want the eat one great! But true turkey hunting is calling the bird up close less than 30 yards is the goal.
Now that being said and it is legal in several states to use a rifle - I stilll would shoot for the head - more of a challenge and no meat is ruined. I think a 270 is far to large a caliber but then again I have never shot one with anything other than a shotgun.
A 22 in the head neck should be all you need. Still be one heck of a challenge with a 22 at some distance.
Be safe
JW