You might want to post this on the tech forum, so that you can get several different and conflicting answers, all vehemently expressed. ;)
Just kidding.
My own two cents (probably worth less than that) would be not to run a high-draw appliance like a microwave using only the batteries -- a very fast drain puts a lot of stress on the components. I would be more inclined to use the generator. But really, this is just my impression, since I have never tried a fast drain on deep cycle batteries. Who knows? Maybe the batteries will enjoy the extra exercise, like a racehorse finally able to gallop. ;)