Yes, the converter (not inverter) should charge the on-board trailer batteries and in your case without an isolater to the tow vehicle it should charge the vehicle battery too.
Make sure of the following:
1. AC power is on inside the trailer (e.g. check a 120V receptacle).
2. The breaker for the converter is not tripped. Turn them all off and back on the be sure.
3. The fuses on the converter aren't blown from a previous overload or reversed battery install.
If you have access to a voltmeter, check that the voltage on the battery rises from 12.x to 13.x or more when you plug in shore power. If it doesn't change when you plug in, the converter isn't powered or working.