Finally! Someone else experiencing the same problem I have been fighting for a year. BAR .308 semi auto has serious grouping / pattern issues. I, like apexerER, finally received this beautiful gun with high hopes only to be let down because of its lack of accuracy. I have not returned it to Browning as you did but sounds like they cannot correct the problem 100% either.
Starting out, 165 grain fusion. 1st 3,maybe 4 shoot in 2-3 inch pattern; not good. depending on, 4th or 5th is bad flyer. I mean off the target bad. Tried various manufacturers, various grains of ammo only to get the same results...all over the place. I have decided that it likes Fusion just not 100% on the best grain it wants to be fed. I am now using 165's but will experiment more with 180's.
To take it a few steps further,I removed the forearm to find that the barrel was not free floated, which I corrected / free floated.
Back to the range, some better, that 4th is always going to fly! Determined, I found the scope rings to be loosening some so I went to a 1 piece DNZ mount. Still not grouping like I expect; very inconsistent but we're gaining ground.
Reluctantly, I went another step and sent back a Zeiss Conquest 3x9x50 ( got to love that lifetime from Zeiss) that I was using on the BAR just to eliminate the scope or rings being the problem. To this day, I still have the gun but have very low confidence in its consistency. Right now the groupings @ 100, using Fusion 165 grain are less than an inch for the majority, BUT, you never really know when that flyer is going to come into play. I guess I can call it satisfactory at this time but I do expect better for 1000 dollars and from a manufacture such as Browning.
I continue to mess around with it but it was immediately put on the bench for a Browning A-bolt SS 30.06 which is definitely a tack driver!
Glad I found your post and glad I was able to share and let you know you are not the only one out there fighting the bar 308.
Regards