If you just come off 12v at the mid post you'll eventually kill the lower battery.
Actually, what happens is you kill the upper battery. In order to charge the lower battery, you overcharge the upper battery. The alternator sees the upper battery as a 12 volt battery with a very low resistance to ground.
sounds a bit hokey - but logically it should work
I wouldn't try it. Logically it might work, but inside that inverter box there are grounds on the output side and grounds in the electronics and I would expect to see smoke.