I am building a mobile sound system powered by a car battery. I have a small mixing table that runs on 12V but it needs AC power (50Hz) and won't work with DC (I tried that, with very weird results.)

Currently I am using a 12V to 230V inverter and the provided power supply to convert it back to 12V alternating current. This seems wasteful and I do not want to introduce another voltage level into my design just to power the mixing board.

How could one build a simple circuit that does this? I have basic electronics skills (soldering, know about resistance, voltage, current) but never built a circuit board. I would prefer to avoid opening the device, except it is very easy to do the modifications.

The mixing board is a Renkforce MX260 USB DJ Mixer. Th manual can be found here.