I dont know about a bb, but it is a 3/16" ball bearing, and may be available from a hardware store like true value, coast to coast, any place that has a big selection of odd fasteners and hardware. (I have got them from our True Value before, for other things) Fowlrocks was gone, and I actually had a drawer full of them and they were from a bicycle neck, they were the right size, so if you have a junk bike laying around, it may be the answer too. (Im good at tearing stuff apart!!) If you have a drill bit index (some times you might get a cheap plastic one in the mail from advertising companys), or a drill bit set that has different size holes for different drill bits, that seem to fit pretty tight, you can try to drop a random ball bearing you find into the holes, starting on the small side, and you want one that wont fit until you get to 3/16". It is a fairly common size, so it shouldn't be a big deal.
If you have no luck, (after atleast trying) let me know, and I could send you one from my drawer.