The A4 barrel will fit as the bearing surfaces are the same, but it is much heavier. That might create problems with cycling in conjunction with the A6 style booster. I have not tried mixing the parts that way, but I probably should. Perhaps someone here has done that and could chime in.
A6 barrels are out there, but more expensive than A4 ones. Figure $100-$150 and they will be in 30-06. If you want to keep your gun in 308, you could have an A4 barrel turned down to A6 dimensions. I think there have been 308 barrels in A6 configuration, but I have never seen one. Don't know if they were commercially made.