On the 2 builds I have with the Bartlein 2b I only had one fluted, its the one in the picture and its a .243. The other one I didn't have fluted - its a 7MM-08. Almost exact same components on each build and they weigh about the same. If I didn't have the .243 fluted it would probably weigh a bit more, if I had to guess the flutes maybe reduced the barrel weight by 3-4 ozs maybe?? Not much I suppose but both rifles balance about the same so I think it did help on the .243. Here is the other one, posted before but likely lost now due to the photobucket fiasco.
2b in the middle, factory Remington sporter on the left, Brux #3 on the right.