If I read it correctly, the slave chips would not have their anodes hooked up at all because they are controlled only by the master chip in your configuration. Said another way, by sharing the led module anodes, the slaves have given up control to the master, otherwise why use that configuration? Don't believe the diodes are the answer here, but a software fix might do it.
Where are the transistors and resistors? Are you multiplexing cathode on times? Looks like the package power rating could easily be exceeded without some of these precautions.