I was under the impression that you wanted to go 6 Barb then the rest fighter, but it looks like you were talking about going 6 levels of fighter and then going back to barb.
In the first case, at best case scenario (a game that goes to 20), you'd be getting 1 extra ASI, and be getting your 5th one level sooner than you would if you stayed pure barb. In the second case, you're getting the exact same number of ASI's as you would if you just stayed barb, but you would get the 3rd one sooner (which would go into CON probably). You do get an additional chance to crit, but it's a pretty small increase (5% to 10%). That chance is increased or decreased based on how many times you can attack and (dis)advantage rolls, but I wouldn't rely too much on advantage rolls (especially when it comes to reckless attacks).
In both cases, we have no idea if Berg would even live long enough to see if those levels sacrificed would be worth it, or if CoS would even be around for that period of time for Berg to get the XP required.