Active Subwoofer Recommendations

A pair of small 8" units, sealed, might work quite well. you don't need much, it may seem loud, but it is still just a sound bar.
I've no idea how the B&W is wired for handling a sub, though.

Rythmik L12 is <$550. It should be good unless your space is huge.

+1 on the Rythmik. Servo Subs.....yum

Paradigm Defiance V10 Retail $550 - Has Anthem Room Correction and full App Control from your phone - wireless capable although doing that is a last resort, better to run a wire - check it out at Paradigm.com and look up videos on YouTube about the Defiance Subs.

3rd the Rythmik L12.  They stand out in sound quality at this price.  The only other subs I would consider at the price point are HSU.