Yes, you need to have your "safety" conductor bonded together. So,like a lot of people say in north american 120 volt wiring, common practice is:
black is "hot" or where the electrons come from
white is "neutral" or where the electrons want to go to
green is "earth" or safety, which is a catch-all for failures
You always "bond" or connect electrically your safety catch-all to the case or surrounding to everything else that is conductive but electrically zero. Various situations call for your earth to be connected, "bonded" to neutral, or isolated, like others have said.
What piqued my interest is nobody addressed your concern about grounding "bonding" both the AC and DC systems together. Which, by the way, is totally fine in your case.
I'll just go through some thoughts, feel free to skip the blather if you already know about this stuff.
120 volt single phase AC power is defined by the voltage potential swinging through zero volts 60 times per second. Peak to peak, "120 volt AC" is actually much higher voltage, approximately 170 volts. The Root Mean Square of the sinusoidal curve as voltage swings back and forth balances out to 120 volts potential.
DC power in vehicles is a constant voltage potential one direction.
Electrical power from batteries is a chemical reaction, specifically a type of oxidization and reduction of different types of metals sandwiched between a conductive medium (the electrolyte). You have to physically stack up your pile of materials to get the electrons to flow.
This means your stack sort of sits there chemically draws electrons from the material until they reach equilibrium (also known as the battery going dead)
This inherently means that battery sources are DC - Direct current.
AC power, like previously described, is mostly derived from rotational generation. Like a big spinning thing. We can go back in time to the industrial revolution and steam power to see the origins of AC, if needed.
Alternating current ALSO originates, technically, from anode and discharges to cathode. However, they are not referred to as such, because they are switching roles periodically (like, 60 times a second). Therefore we can just refer to them as "electrodes" aka "the place that electrons come from"
In all cases, DC power and AC power can be derived from physical movement, through a magnetic mass (like iron) and inducted into a conductor (like copper) which pushes electrons out the end of the conductor.
Folks realized there were a lot of benefits to Alternating Current vs Direct Current. From safety all the way to the amount of voltage that can safely be distributed via AC.
We won't get into many details about that, but one of the prime safety benefits of AC is that it is self-extinguishing in the case of a fault. A DC arc will continue to persist as long as power exists. An AC arc self-extinguishes 60 times a second, due to it's voltage potential crossing zero 60 times a second.
Basically, you get a big BANG with DC and an big BUZZ with AC.
Work with electricity only happens when you have a potential between your anode and cathode, or between your electrodes. It doesn't matter if it's cycling 60 times per second or constant.
Therefore, when you are "creating" AC power from an inverter or a generator, there is always a "base" or reference voltage of which the relative potential from it to everything around it should be zero volts.
The battery anode, (the negative post) reference is zero. The energy converted to alternating current uses the same zero reference. Generators that may be attached to the same system use the zero reference in the electrical coils.
This is the reason that your inverter or generator generally maintains the only "bond" between your earth and neutral. The alternating current electrons being shuffled back and forth through those devices use the "earth" as the reference for "ground". Without that reference, you get a potential difference, and therefore the possibility of electrons flowing between earth and neutral, and therefore shock hazards.
When power originates from somewhere else (like grid power) the earth to neutral reference is maintained in the utility itself. Adding secondary bonds between earth and ground in those systems can actually add a potential hazard (shock hazard) when an inequality exists, such as consuming some of that energy in a motor, light, or other device.
If you've read this far, the summary is this: internally originated AC electrical sources are friends with internally originated DC electrical sources, because they generally share the same reference voltage. There are of course many exceptions for this, but in this case they are completely compatible and actually necessary.
