To produce ABs in cell culture you usually create a hybridoma cell line, by fusing the B-Cell to an immortal tumour cell line (as you postulated). This approach does however require you to identify the cells that produce the proper antibody.

It gets much easier if you know the exact sequence/structure. To learn this (or find a better one) it is also possible to generate libraries of antibody sequences with semi-random hypervariable regions (the part that binds the antibody) using for example phages that will display any protein on their surface area and then measuring the binding affinity! This is called - drum roll - phage display (very creative).

Alright, enough trivia for now.