12V DC to 12V AC (50 Hz) converter / inverter for low loads (< 1A)
12V AC has a 17V peak, on either side of neutral. So that is 34V peak to peak. The simplest way to get a sine wave is, ironically, using an audio power amplifier. Feed in a 50Hz sine wave signal, and adjust the volume control till it reads the required output voltage. The power supply for the power amplifier may need to be +-19V (i.e. 38V). This may be difficult to do with a standard boost controller since it is close to the upper limit of most boost supplies. Additionally, the neutral line will be shifted up from 0 VDC, and it is the neutral which will be 0 VDC from the output of your sound system. This will be awkward.

It is probably best to open it up and check the actual voltage after the rectifier, and to supply that from your battery via a boost converter (if needed) and a properly rated diode. (Using a separate diode will save the rectifier diodes, since they are likely rated for lower current.) You can also check the peak voltage on the signal path, and the headroom required for the parts in the mixer box. It may be that no boost converter is needed, if the signals are all under 9V.

added

Some more light for the photos would have been better. But, the 7912 is visible. So, this is a differential power supply, and requires -12V. So I guess you are out of luck. What do the other ICs say? It may potentially be possible to use two 12V batteries to power this thing, but one may need extra low drop diodes to protect the voltage regulators. Not trivial in the amount of space in the box.