The displays I'm using are both very difficult to find and rather expensive, so I'd prefer not to just toss the hardware I have now.
Most of the other options for switching the outputs I've found would either require some reflow soldering (which I can't do) or wiring up dual quad-SPDT analog switches for each display (which would not do much for my component count).
I do have some MAX6969 chips (Mixed-signal and digital signal processing ICs | Analog Devices) which would do exactly what I need to do and only require 9 chips for the whole display array. I think at this point they're the better option than trying to jury-rig three 7219 chips.
Thanks for the suggestions!