I would hold off buying a new battery for now. Certainly pick up a replacement BMM. This newer revision from ebay seems good value, assuming it's undamaged, the seller will accept returns.
http://www.ebay.co.uk/itm/161819882804
It is a different shape to our existing modules, but the top left mounting is in the same location so I hope there's no complications with fitting.
One thing I noticed from your VCDS scan is that the Shop# is different on your BMM module compared to the majority of your other modules. So maybe the previous owner had problems and had it changed.