I don't have a B&R bag, but I think you might be looking for an umbrella with a flat handle. I have a manual "Raines by Totes" umbrella that I travel with precisely because it's small and flat when closed. The style is labelled "RSTCS," but unfortunately, I haven't been able to find a link to it on the Web (which is a shame, because I would buy it again if/when it needs replacement).
The closest I found was the
Samsonite Manual Flat Compact Umbrella. You might wish to check its advertised dimensions to see if it fits the pocket on your B&R.