I would think that connecting the ground of the AC to the ground / negative of the battery would be no problem. With the rubber tires, that ground is floating wrt to Earth, I imagine, so connecting the AC ground (real earth) to the battery - would just make the battery - voltage down to Earth...
personally, I'd rather have a separate charger. More versatile, I think.
coming from a US perspective on wire colors, I must say I was surprised when I saw the colors used in the EU, with brown and purple or something like that. Seemed like lacking a natural association with cold / warm etc...
well, an inverter is meant to go from DC to AC so this thing will run backwards as well like a charger?
As to wiring, I don't think your colors are standard for AC if that's what you meant.
Maybe best to get a multimeter and do some reading first?