A "converter" is typically a DC power source that charges the batteries, supplies DC power to the house loads and has a built in DC power distribution center, with fuses and connection points for the DC loads. Usually it also has the AC main breaker and the AC load breakers for the 120 VAC circuits. While an inverter often has the battery charger and can supply DC for the house loads while plugged in and charging, it really never has the rest of what a converter has, it needs to have a DC distribution fuse block built separately, and the AC breaker box also separate. So no, I am unaware of any single unit that encompasses all of those functions.
A lot of converters are biased towards supplying a constant voltage supply to the house loads, I have one that always supplies 13.6 volts for example. That means that it is not as good a battery charger as one that does a higher voltage bulk and absorption charge cycle. An inverter with a charger, particularly the higher end ones, will have a three or four stage charger that delivers the varying voltage charge that is better for batteries, but can cause issues for some DC loads that expect a constant 13 - 14 volts all the time.
Brian
2005 F250 Supercab, Powerstroke, 5 speed automatic, 3.73 gears.
20 ft race car hauler, Lola T440 Formula Ford, NTM MK4 Sports Racer
1980 MCI MC-5C highway coach conversion
2004 Travelhawk 8' Truck Camper