If you need to "translate" a matrix array of buttons (typically a numeric pad) to a DIFFERENT array of buttons, then yes, an Arduino (or Arduino-like) microcontroller is a good solution. It would be easy enough to:
1) Create a matrix of relays in the same pattern as the 3189 machine
2) Program a microcontroller to recognize all 12 combinations from the 3500 machine keypad
3) Have the microcontroller activate the 3189 relay matrix pattern with the equivalent button push.
Arduino-like boards are available for <$5. My favorite is the "Pro Micro" version
You may have to use "expander chips" to get enough inputs and outputs.
The favorite input expander for Arduino is CD4021 parallel in, serial out (8 channels)
And the favorite output expander is the 74HC595 serial in, parallel out shift (8 channels)
You can get off-the-shelf solutions for both input and relay output expansion:
16 channel relay output board:
https://smile.amazon.com/dp/B0057OC66U16 channel input/output expansion board:
https://www.amazon.com/dp/B07GRQWQF9