Schematic & pin reduction
Use another multiplexer to read the 8 ID switches, selected by the same 3 bits as the others and with its output going to a single digital I/O pin. That should save 7 pins, leaving 3 spare for other things that you haven't thought of yet. 

You could use a digital multiplexer such as MC74HC151 or 74HC251, but it might be cheaper and easier to just use another HCT4051, as then you only need a single pullup resistor on the digital I/O pin, rather than 8 (one per switch) on the multiplexer inputs.