Hi - to add my two pen'orth to the discussion, adding BBC Sounds as an app to the B2 would be a great addition, but there are a couple of work arounds you could use instead.
1) Open the BBC sounds app on your smart phone or tablet and then switch the B2 to its Bluetooth inputand pair your deveice with the B2 . However, Bluetooth has only a very short physical transmission range and it doesn't pass through solid walls or even furniture. Ideally the B2 needs to be within 20 feet line of sight for Bluetooth to function.
2) Buy a Chromecast Audio - a small £30 Google device ( now no longer in production - but still available on a well known on line auction site ) and plug it into the B2 Aux input . The stream signals via WiFI from the Smart deveice to the B2 . The audio qaulity will be a lot better with full frequency response and wiFI has a much better transmission range and penetration than Bluetooth. If you have a digital amp the Chromecast Audio has an optical output
On a separate point : the B2 has a good optical didgital output, but no optical input.
Might this be a worthwhile future development if the capacity to do this exists in the Raspberry Pi?