Hello and welcome. First, your post is asking a question about CONVERTERS, not "inverters". They are two different devices - an "inverter" changes 12V DC electricity to AC (Alternating Current) or, "house power" 120V electricity. A "converter" produces 12V DC power, in our trailer it changes the 120V AC shore power to 12V DC but converters can also reduce high voltage DC power to 12V DC which they do in a Hybrid automobile. I'm not being pedantic, but just want to avoid misunderstandings in what we are talking about.
Now, the converter that is charging the battery is changing your shore power input to 12V DC and that will power the heater, lights, fans, etc. and will charge the battery. It actually delivers something above 12V to charge the battery and that can be as high as 14V. You can tell if your battery is being charged by placing the probes of a multimeter on the terminals of the battery and checking the voltage. It it's above 13V, the converter is charging the battery. The charging voltage will be close to 14V if the battery charge state is low, but it will taper off to close to 13V when the battery is fully charged.
The Air Conditioner (A/C) is powered by 120V AC power, and the converter DOES NOT power it! The converter has a bypass circuit inside that simply sends the shore power unchanged to the A/C and it can also deliver shore power to a microwave, the AC function of the refrigerator and an AC heater element in the water heater using that same bypass circuit.
There is no reason that you would have a second converter in your trailer. The converter he put in the cupboard would do everything you are talking about.
However, the other device could very well be an INVERTER. It would be there to give you easy access to 120V to power something like a coffee maker, power a laptop, hair dryer, or something similar using input power from the 12V battery. An inverter will always have a standard looking power plug socket on it where you could plug in one of those household items.
So, I've told you how to determine if your battery is being charged and how to tell if the other thing is an inverter. But if it's actually another converter, I don't know why it would be there. Is it possible that it is a "charge controller" that is connected to the solar panel you mentioned? If it is, it would have wires from the solar panel and another connection to the battery, with nothing else.
- Jack