There should be no ground currents unless there is a fault condition. If ground and neutral are bonded at both ends then the main current will flow in all three conductors, if the neutral current is compared to the live wire current it will no longer be in balance and trip the protection.
You have an unenviable situation where you have two (perhaps 3 if you get grid connected) power utilities on site with the generator and the battery powered inverter. The responsibility of providing the neutral conductor at (close as possible to) ground potential rests with the supply utility (which is easy when there is only one).
You mention the 230V input on the inverter unit. Twice in relation to battery charging and one in relation to synchronising. These two do not work together often. If you have a battery charger input on the inverter unit then it does not need synchronising. If the 230V input is for hot switching between the inverter and the generator (or utility grid) it is not just for battery charging and is the sign of a complex or sophisticated controller.
There are likely standard connection guidelines for this controller that you should investigate, if you can locate an on-line installation manual this would help with getting answers.
Also as mentioned you should try and maintain a single ground point at your main distribution panel (unless you have a very large installation with possible isolation transformers where you have to re-establish the neutral ground reference after the isolation transformer).
The inverters 230V input neutral connection should NOT be tied directly to ground as it is not a supply neutral but it is possible that it is connected to the output neutral if it is intended for hot switching and not just battery charging. If this is the case you may be able to float the generator output (unlink the ground from neutral). You may have to remove the RCD on the generator if no other solution is found. I found an interesting document that discusses the safety of these options though based in Australia. There is also an Australian forum thread that covers and solves what is possibly exactly your problem.
Basically if your generator is only supplying your inverter/charger/controller it does not need the RCD which is used for protecting humans from earth faults. The RCD on the inverters output will provide this protection.
EDIT:
The on-line installation manual has a lot of meat in it. As predicted the unit is sophisticated and does support battery charging and hot switching. Specifically the main guide describes the functional parts of the wiring diagrams in section 17 Figure Elements AC Part and the appendix shows the wiring options. All of the wiring options (except the transformer isolated (label Y) one where it is required to re-establish the neutral reference) do not want the neutral connected to ground (label C) at the inverter/controller. The neutral line is illustrated as linked through (which is acceptable as it is also able to switch to operate in bypass mode) this means any neutral to earth (label C) link will be seen by the output of the generating set.
What is even more interesting is that the neutral can be programmatically connected to ground inside the inverter (label V or relay K4) but only with expert level parameter settings according to local codes and specific needs that would be rare.
It also mentions the key point "The neutral is earthed at a single point of the installation, downstream of the source and upstream of the protection device(s)" (with some extra details about multiple sources again) but it boils down to the fact that you cannot connect the neutral to earth AFTER (they say downstream) the RCD. The only time you would use the link at the inverter (label C) or the internal expert level relay (label V) would be if the inverter is the only source of supply and has to take charge of the neutral reference. The reason to link at the inverter programmatically would be if the neutral reference is supplied by a plug in generator or shore power and this reference is lost when unplugged.
Conclusion
Your generator RCD trips when the inverter has synchronised and connects the load to the generator, at this point the current shared by the neutral and ground is no longer zero and the unbalance occurs.
You can leave the generator RCD (label D) in place and the generator neutral to ground link (label E) but will need to make sure there are no other neutral to ground links in your system (label C and V). For peace of mind also document this so if the generator is disconnected (label K) a neutral to ground link must then be established at the inverter (label C).
As some highlander is rumoured to have said; "There can be only one"