Where do B cells produce antibodies?
The oncologist seems to be incorrect. 

Antibodies are created all over the body inside secondary lymphatic organs (lymphatic nodes) or any other mucosal associated lymphatic tissue (MALT). 

B cells mature in the bone marrow and finish their maturity in the periphery, then they can live in lymph nodes, MALT, spleen, omentum or other structures and they can create antibodies in all of those structures. 

In regard of your localized infection question, you are right. When a pathogen enters the body the B cells closer to it are in the lymph node that is closer to that site and thus there is where the antibody response will take place. That is why when you have a sore throat (throat infection) your tonsils are swollen (they are lymphatic organs too).