You could use the single output pin method driving 3 cascaded CD4026. That is, you have a 3-digit decade counter that counts 000...999 by using the carry-out pin to drive the clock of the next counter. Can get a bit messy since changing just the left-most digit involves sending 100 (to increment) or 900 (to decrement) clock pulses! Changing just the right-most digit (without affecting the other digits) can get interesting too! But it does meet the single-output pin criteria. Obviously you could set up the 3 displays as 3 independent CD4026s but that would require 3 output pins.
So in this example the shift-register method might make more sense. That is you cascade 3 74AC164 serial-in, parallel-out registers to create a 24-bit output. You always need to send exactly 24 Morse-code pulses to change any display. But it does meet the single-output pin criteria.
To your point, if the box is 10 feet away, having a separate Arduino becomes attractive. As I recall there was angst in reliably decoding the analog voltage from the potentiometer using an A/D input of the Arduino. Having that analog voltage run 10 feet down a cable would not help matters!