Hi Shaun, in my opinion the 4B3 driven by the BR-20 analog preamp section plays in another league and cannot be compared with a digital volume control which is more a workaround not designed for the best audio quality. I own a 4B3 + BR-20 combo and I can tell you they sound in an incredible way.
Additionally, the BR-20 embedded streaming computer is slightly more powerful than the one embedded into the BDA-3.14 but this will not affect the audio quality.
As a cost-effective alternative, why don't you try adding a second-hand BP-17 or BP-26 preamplifier to your system?