I'd try both Federal Power Shok soft points and Federal Fusions. I have a older BAR. The ones before the Safari ones. I tried shooting Winchester Ballistic Silver Tips in it and it was all over the target. A 6" group would have probably been a good group for them. I got frustrated but tried some others. Remington Core Lokts didn't group much better but a little maybe. I had a 8" target and the shots were all over it. I then tried some Winchester Soft Points which were slightly better but still not great. I then tried Federal Fusions and my groups shrunk down to about 1.5"-2". Not that's not great but it's a lot better than being all over a 8" target.
I later tried Federal Power Shoks and they seem to have either the same or very close to same POI as the Fusion and I get almost as good of groups with them. The first time I shot them I got a 3" group where I got a 2" one with the Fusions. However, at $14 a box vs $30 a box I thought that was fine for hunting. When I shot it this year I shot a 5 or 6 shot group with a flier or two I can't remember and the rest were touching. I've gotten several groups that were within 2" with the Power Shoks. I've stuck with them.
My experience with these guns has been that they just aren't that accurate. I mean I like them a lot for hunting and for hunting they are plenty accurate but I'm not sure they are a moa gun. At least mine doesn't seem to be. I've read reports where guys on here claim theirs are moa or sub moa guns but I've also read quite a few like mine where guys say 2" groups or so is about the best they can do with factory loads.