can’t rely on a genset that small having anything remotely like a conventional TNS earth, it will be as floaty as a cloud.
Maybe, maybe not ...
So you will have to make the N-E link yourself.
And possibly remove the earth from the centre tap of one of the windings. Not uncommon to find that there is 55V N-E and 165V L-E. Two 110V windings, parallel for 110V, with a winding CT to earth to make 55-0-55, and it ends up at 55-0-165 when the windings are put in series for 220V.
As discussed in other threads (here and elsewhere), you have a fundamental problem if you don't hardwire the genny. If you hardwire it, you can make sure the correct N-E linkage is made so that circuit protection will work(ish).
But if you provide a power inlet, then you have no control over what someone may connect to it. You can't put a N-E link on the fixed side as that would cause a short if one of those 55-0-165 gennys were plugged in - and I would bet that a lot of them don't have double pole protection so wouldn't trip due to this. But you can't not provide it as otherwise there isn't a N-E link if the genny is fully floating.
Personally I think there's an argument for treating it as if it could be an IT supply - as long as all the bonding is in place, all it means is that nothing will trip on a first fault, but without removing covers, that first fault won't actually create a dangerous voltage that can be touched.
first you will have to split the circuits you want backup power to.
I wouldn't bother. The householder will quickly work out that a small genny will collapse if they try using the shower, or the kettle at the same time as other things, and so on. Even if you split circuits, it'll not stop someone moving the (e.g.) kettle to a genny backed circuit. And however you decide to split them, it will always be wrong sooner or later.
Simpler to just do the whole installation and educate the householder in turning stuff off.
And, a 3kW portable genny is likely to struggle to trip most of the OCP devices anyway. It won't trip a B32 on a socket circuit, and it may or may not eventually trip a B6 on a lighting circuit. So most of the protection is going to come down to the built in protection in the genny.