Not an expert, but as I understand it the purpose of the ground is to give a path for the hot wire such that if it comes into contact with anything it'll immediately short (thus tripping the hot wire's fuse). So, as the power is coming from the inverter and it probably has a circuit breaker (or it's the next thing inline), then everything should be grounded to the inverter. I suspect the negative terminal on the battery should be grounded to it as well (although the inverter manufacture may have already done that for you).
The tricky part, and one you probably need an answer from someone more knowledgable, is what to do with the ground from the mobile system if powering something that has a different ground; that is you're powering a house from your mobile platform. If your AC fuse is 60 amps on the mobile system, would a 15 amp fuse in the house blow on short thus protecting the house? I believe my generator is the same scenario, and when connected the grounds are connected which I believe provides a full round-trip path and would allow the house breakers to trip if needed.
Anyway, something to get you started perhaps (or for a real expert to point out why that's all wrong ?).