You might just need fatter wires between the inverter and battery.
If you plug the inverter into a 12v socket then that leads to long thin wires which is the problem. If it also comes with a set of wires for battery clamps then use those instead of the 12v plug set directly on the battery. Or if you need to do this inside the rig, you can clamp to the battery lugs on the DC panel which also has the converter's wires to it. The wires from the lugs to the battery will be fatter than from the 12v socket (on the Winegard panel, eg) and might make the difference.
Getting a bigger inverter won't help with this.
