As the owner of 4 BPS's, 10 ga. and 2 3.5" 12ga. guns, I have seen what you are experiencing, and it is likely one or both of two things. (1) If the shell is failing to feed from the magazine, it may be the fault of short cycling the slide. Be sure to pull the slide all the way back. There's a little tab on the bottom of the right side shell stop (as viewed with the action upside down). This tab must be depressed by a cam surface on the action bar far enough to release the shell in the magazine. That requires you to pull the slide all the way back.
(2) The second thing which may cause it, and was the cause of a problem in one of my 10ga. BPS's, is the shell stop may be projecting a little bit too far in toward the center of the action to release the shell. I had the same problem with one batch of Federal hulls, while it worked fine with Remington's and another batch of Federals. There's a wide variation in shotshell tolerances, from brand to brand.
I filed off about .005" from the tip of the shell stop and that cured the problem. It was long enough to stop the Remingtons from feeding prematurely, but short enough to allow the larger problem Federals to feed when I worked the action. I found there was a .015" difference in rim diameters between the Remingtons and problem Federals.
If changing shells doesn't fix the problem, you will need either to take it to a competent gunsmith or send it back to Browning with the problem shells, so they know how much needs to be removed for it to function properly.