Yes, you can run the VL2 thru a mixing desk and then connect to your computer. I'm not sure why you want to use the VL2 as a vocal processor in BiaB though. You might consider using RealBand or any other DAW for recording vocals and other tracks you don't use from BiaB.

Could you tell more about what you want to achieve with the VL2 and the mixing desk? That way I can give you a better answer to your question.