If you connected one inverter to each 12V battery (assuming you're using 2x 12V batts in series?) it could work,IF: The loads on each inverter were equal(to keep the batteries loaded equally),and the 120V circuits were completely isolated from eachother,and probably isolated from earth/safety ground too,atleast for one of the two 120V circuits/inverters.But that also depends on the way the inverter is grounded..If the 120V side ground is completely isolated from the 12V input negative terminal,it should be fine..But,If the 120V ground and 12V neg. terminals are connected together(inside the inverter),that would be a (smokey) problem,for atleast one inverter.
(It would cause a ground potential difference of ~12V,since the ground of the inverter connected to the "top" battery would actually be floating ~12V(= "bottom" batt.voltage) above true earth/safety ground.