DC to DC charger

Solar availability varies a lot by area. In Western WA/OR, it’s basically unavailable for for most hours for months. ;) Solar doesn’t work very well in snow or heavy rain either.

It’s a good idea to have a DC charger for backup assuming you have the money to spend.

Charging is separate from running an inverter. So, yes, you can use a charger and also an inverter.

Think of it this way.

You are trying to fill a water tank (charge the batteries to 100%).

You are adding water at the top (power from the DC-DC charger).

Some one occasionally opens a valve at the bottom for a few minutes (inverter must run the refrigerator).

As long as the tank never runs completely empty (batteries below the level to run the inverter) there are no problems.

Solar is rarely completely unavailable during daytime hours. Is a 120 volt fridge worth all this trouble? IMO no. However, adequate batteries and panels would solve your issue.

The DC-DC charger charging the TC battery and the 2000w inverter powering the fridge would operate independently.
YES, both can be used at the same time.

Assuming the refrigerator doesn't draw more than the 2000W and some kind of battery in the loop. A DC-DC charger of min. 20 amps off the alternator should have some excess charge.

Best addition I ever did to my truck camper. I've got a lot of solar/Victron multiples and MPPT controller, but the DC to DC charger took out the worry of topping off the batteries on a cloudy or rainy day.

My particular unit puts out 30 amps.

You won't regret it.