Two 6 volts is definitely not enough. If you try running a 700w microwave, for example, it will at least set off the inverters low voltage alarm... and it will cause a shut down if the batts are not near full. If you're thinking, 'well, I don't intend on using such a large load', then you also do not need a 2000w inverter.
I'm running my 1000w xantrex off four 6's. On two near full 6's, the alarm sounded when running the 700w microwave. It was not an inverter issue, but a battery capcity issue. The inverter did what's it was designed to do.