The unit may not be able to charge a completely dead battery.
Little confused! Your rig should have a converter. Yes, it is possible that it has a converter charger installed but would be higher wattage, 1500 or 2000 watt.
An inverter is good while dry camping to supply 120 AC volts for items like TV, DVD/VCR and computers. A 1000 watt would handle these items easily. It would not operate larger items like air conditioner and many microwaves.
A good replacement converter would be Progressive Dynamics 9200 series. I would be looking at a 45 amp 9245