Whichever you shoot better but...the 06 is giving you weight and frontal area.. Echoing the what others have said, if you need to use a 180, at least at shorter distances, a 180 partition will likely leave a larger exit than a 180 barnes (if you are close enough to get the partition to fragment some). The 150 barnes is a better choice than the 180's for you. They are relatively long bullets for their weight so also might shoot well in your rifle.
Only killed a few black bears and they weren't hard to kill, but as mentioned they all were kind of hard to find afterwards. One died in a blowdown, less than 100 yards from where it was shot...took hours to find because of all the shadows etc.
