Driving 16 Common Cathode Displays

Yes, you are correct. Ht16k33 drives 8 cathodes and 16 anodes. If you can't organise your displays like that, and it sounds like you can't, it was not a helpful suggestion.

Ok, so... 16 cathodes and 8 anodes...

Do you have 11 Arduino pins available? Use 8 pins to drive the anodes. The other 3 drive 2x tpic6c595 shift registers (daisy-chained). The Arduino must do the multiplexing in this circuit.