Yep I'm with Nighthawk on this one. Modified sine wave inverters are getting so cheap it's almost a waste
not having one! However that's only true up to around 500W. If you need lots of power then inverters get expensive and generators start to look more attractive. Also if you fit a 2kW inverter and try to run it from a 'normal' size battery, say 100Ah, it'll probably only last 15 minutes

at full load.
As for your control box, twin batteries can be arranged differently deppending on what they were used for. Some are just wired in parallel to get more starting power on cold mornings. Others, as yours seems to be, are wired as seperate batteries, one for starting the vehicle and the other for running equipment when the engine is stopped. The batteries are disconected from each other when the engine isn't running so that you can't flatten the starter battery with your lights/inverter or whatever and you can't try to start from the auxilliary battery which may not be rated for starting current. It sounds like your box is such a seperating device. Normally this is done with either a simple relay, large enough for the current, or a set of diodes. More complex and expensive systems exist though.
Steve