I ordered the Crosstac AICS vise block from above and it came in today. Just handling it out of the box and fooling around with it I think it's just what the doctor ordered. Per the description there's four adjustment screws that allow you to get the block fit properly to the magazine well so there's little to no slop. My AX AICS required the most amount of adjustment because of its funneled shaped magwell but I was still able to get it on there snug so that it didn't rock back and forth. There's no lip on the backside to catch the magazine release lever, it's all friction fit, but I don't really see that as being a necessity to be honest. I tried the mag block on my AX AICS, an Oryx chassis, and a factory Bergara HMR and it worked with all of them, albeit they all required their own little tweaks to the adjustment screws. All in all I think it was money well spent.
They make a version for AICS/Vudoo, however I do not know if it would be compatible with the Bergara B-14R, may have to email them about that.