What appliances are you going to plug into the "NEMA 5-15 sockets" and where will they be? GFCI protection is not required because something is an appliance. Other things are considered.
You state you have a portable box, this is for a specific use? Back up power when the grid goes down? Could it ever be used somewhere else?
Can you give a me a link to the manual for your specific model Inverter charger? I would like to read about the bonding in the 2 different modes.
There are ground fault circuit breakers that fit into standard circuit breaker panels. You could get a 2 circuit panel that would accept the GFCI breakers and then wire to a standard duplex receptacle. You could even break off the tabs on the duplex receptacle and wire the 2 circuits to it.
What you have asked about is a very good ,and interesting question, "My question is: Do I need GFCI/RCD devices (or integrated into the sockets) in addition to the breakers? If so, why?" With more information I can give you my opinion on what may be required. You are getting in to gray areas of the NEC. Have to look at requirements and intents of the codes and what will be safe for you.
As solar generators are now available, and portable batteries with inverters too, what will be required by the NEC. Does it really matter how portable 120 volt power is generated. Gasoline, diesel, solar, storage batteries with inverters are doing the same thing, producing 120 volts are when grid power is not available or wanted.
Portable generators use
2014- 2020 NEC. 590.6(A) (3) All 125Volt or 125V/250V, 15A, 20A and 30A receptacles outlets that are part of a 15kw or less portable generator shall have.... GFCI protection. . This does get into gray areas of the NEC with new portable batteries and inverters, solar generators that can replace conventional gas generators. This code address the electrical output of something that is portable and generating 120 volts of electricity. It can be concluded that solar generators and a "portable box" are covered under this code.
Temporary power used by personal in construction, repair or similar actives, is also required to have GFCI.
GFIC's monitor the current on the neutral and hot wire. If it varies by more the 5ma the power coming out of th GFCI is shut off. On circuits that have GFCI protection, if someone were to grab the hot and neutral wires, the GFCI would NOT trip, as the current (amps) is still the same on the hot and neutral. Dropping a 2 wire extension cord into anything that is not grounded, such as a sink, that has plastic pipe running to it, an above ground pool, ect, again it will not trip the GFCI, as the current on the 2 conductors will continue to be the same. For a GFCI to trip some of the current must travel through a ground ( a ground fault). A neutral wire can have a ground fault too. So if the extension cord has a grounding conductor (3wire) or the water container is grounded in some way the GFCI will trip. This is one of the reasons pools need to have the water bonded to ground. A GFIC does not need to be grounded, to provide some protection. As there can be many ground fault paths. Such as copper plumbing, damp concert basement floors. metal HVAC duct work, metal gas pipes. ect. In the NEC there has been a provision for installing GFCI receptacles, on 2 wire wiring ( with no grounding conductor)in very old homes.
