bi-directional transcription experiment
How well is gene A annotated? Do you have the gene A sequence (after post-transcriptional modifications)? If you do, you can order it from a company like DNA2.0 and they will synthesize it for you for like $0.35/base. Then you can transform the cells with this plasmid and do a knockout of gene B by inserting some sequence at the ~200bases overlap.

Also, I guess you can transform the immediate transcript (without modifications). It will also be modified once it is in the cell. But it will be longer and more expensive to synthesize, but there are a few methods you can use to do that. 

Let me know if this idea appeals to you. I can give you more info if you are interested.