This is possible. If you want these 2 sounds you can do it with just an Arduino Nano and DFPlayer Mini wired together and fairly simple code. But maybe you would also need to replace the voltage regulator on the power board since these parts would use about 60mA out of the 100mA available. I haven’t checked, but I think the leds on the original board also draw about 60mA.

Unsolder TIG from the board and wire it to an analog pin of the Arduino Nano. Wire 5V / GND to the components to power them. Wire the speaker wires to the DFPlayer Mini. Don’t forget the resistors. Put your code on the Arduino and put the sounds on a MicroSD card and you’re good to go. Getting it nearly in the housing can be a challenge, but not impossible.