Depending on what type of battery charger you have, the automatic changeover from mains to generator might be easy, and no actual switching device needed.
If it is a switch mode battery charger, it will work like this:-
Main in -> rectifier -> 350 V dc.
350 V dc -> switch -> 350V pulses at 20 kHz ish
50V pulses at 20 kHz ish -> transformer -> 14V pulses at 20kHz ish
14V pulses at 20kHz ish - > rectifier -> 14V DC
If your battery charger is like that, you can buy a couple of rectifiers rated at 600V and 20 A (
MULTICOMP|GBPC2506|BRIDGE RECTIFIER, 25A, 600V | Farnell UK). Connect the AC side of each to the mains and the generator, and the DC sides in parallel and connect to the input of the battery charger.
This will not work if you have a linear battery charger. Tell us the make and model or the rating and the size and weight and we can probably work out what type it is.
(A linear battery charger needs a transformer that runs and 50 Hz. A switch mode battery charger needs a transformer that runs at 20 kHz, and if far smaller and lighter)
Some other comments:-
The scheme looks OK. Everything looks possible.
I would charge the batteries way before 10V. By the time lead-acid batteries are down to 12V with a light load they are about to drop.
You need a separate battery for starting the generator.
You need something to decide when to stop charging. Time might be good as you don't want to wait until the batteries are full, as you might have the sun come out or the mains come back on, either of which is cheaper than the generator to run.
If your mains load is big, you might want to bypass the charger - battery - inverter bit when the mains or the generator is working. The charger will be 80% efficient, the inverter will be 80% efficient so the total will be about 64% efficient. (the range could be 40 - 80%. I have made wild guesses on the efficiency of the kit). If you bypass the charger - battery - inverter bit you would save maybe 30% of your power from the mains or the generator.
Whether a bypass is worthwhile depends on how much power you use at 240 V, how much time you run from the mains or the generator, and how much electricity costs and the cost of extra fuel to drive the charger and inverter losses when the generator is running.