The problem lies with the floating earth arrangement, on normal generators there will be a winding N-E, connection, which is not the case in this instance. All you are doing with the earth electrode you have installed is earthed the frame. It can be adapted, but you would need to know what you're doing. Is it really worth it??
To be honest, if all you want from this generator is to power up a socket(s) and a couple of lights in a garage, just leave the thing as it is, it should be fine for electric hand tool etc, just not anything that has sensitive electronics.
I bet if you look in the manual, it will have a disclaimer of some sort or other stating that the genset is not suitable to power household installations or appliances...
If there is no mains supply to the garage you could go through the Garage CU via a suitable plug and socket arrangement. (female plugtop on genset lead). At least it will then still be portable...