Actually, Strikes and Spares has four chimes.

Because the chimes are driven by solenoid driver board pulling the coil output to ground, connecting a logic level sound board is quite simple, just connect the sound board ground to game ground, and connect the chime outputs to sound board inputs, leaving the +43V coil common wire disconnected. Maybe you need pullup resistors at the sound board inputs, to +5V or whatever the sound board voltage is.

The (+) wires are all connected together and that goes to +43V. Do not connect that to soundboard.
The (-) wires are pulled to ground when MPU wants the chime to operate. So you should have an active-low input on the mp3 board.

The board looks very suitable. Connect the sound board "GND" to game ground (NOT to +43V!!!). The chime wires go directly to K1-K4, no extra resistors needed. Disconnect the wires from chime, do not leave the coils connected.

For the 9-24VDC input, I recommend you use a separate small power supply for better quality.

Yes the +43V is not needed. Good luck with the project!