Guidance on programming control registers in a device (DAC)?
USB adapters exist to access I2C devices. Both professional tools and cheap hobbyist gadgets. Or you can make one yourself with Arduino or Raspberry Pi. But you must check if the MCU is still controlling the DAC or not, because the MCU might not expect other masters on the bus. It might just be a fail how it reads the buttons.

The registers just have bits or numbers for controlling the chip. Since you want to control volume, you would want to access the volume registers, and see the detailed deacription what to write there to get the volume you want.