Replace mains 12v power-supply with 12v battery
The power supply looks like a regulated 12V switchmode power supply, therefore it should output 12V under varying load conditions. The battery is not regulated, a charged battery can have more than 13V and the voltage drops when discharging the battery. If the piano needs regulated 12V then connecting to a battery is a bad idea. It is also possible that the power supply is unregulated so the piano might work with the battery. If you have the service manual of the piano maybe a safe voltage range can be determined. Perhaps a range reads on piano 12VDC input connector. It is not even clear how much current the piano needs, it may read on the piano or manual. If you do try this, at least put a fuse (2 Amps?) right at the battery terminal to protect people from burning wires and exploding batteries.