You need to find out what model converter your unit has and find the manual online. That will have a troubleshooting guide. The converter takes 120V and converts it to 12V for all the 12V systems in your RV. It also charges the batteries.
An Inverter takes 12V, when you are off 120V and on 12V batteries, and converts that to 120 to run things like your microwave or TV. It has nothing to do with your trouble.
If you are plugged into shore power then the converter is providing the 12V not the batteries. Get a Digital Multimeter and start checking voltages.
