Sorry Jill but the answer is “it depends”.

The correct size depends on your power budget. Your stipulated conditions is pretty much what my wife’s uncle has to deal with. Six months of winter during which we get virtually no sun and the mountain ranges shelter the homestead from good winds. Mind you when it blows in winter; whooo boy, look out! Today the wind chill factor was -47C cold enough to freeze exposed skin in a couple of minutes. Stupid me forgot my gloves and nearly froze just crossing the parking lot.

His whole house runs on a 3000 wat Outback inverter BUT the deep submersible pump needs 7500 watts to turn over and pump water so he ended up getting a 10,000 watt diesel generator.

Ironically If he had listened to some good advice the demand would have been a lot less.

I picked up a new single cylinder diesel from a place similar to Harbor Freight and bolted a 100 amp alternator to it. This unit uses about one quarter of the fuel compared to his 10 kilowat genrator yet charges the battery faster because it uses a more efficient charge approach.

If you can afford it, the Honda EU 2000 would be a good choice. It can drive a 1000 amp charging source at full output yet when the charge tapers off, the automatic throttle slows down the engine and thus uses less fuel. I have come to the conclusion that gasoline generators are a better choice for cold weather starting. Trying to get a diesel started in freezzing temperatures is a major effort. My gasoline sno blower (twice the horse power)is a breeze to start.

Noise is highly subjective. What is no problem for one person is enough to drive somebody else round the bend or up a wall. Be aware that better mufflers can be installed on just about any motor. It may require a bit of welding of the joiner pipe. I once had to contrive a totally silent generator for a power utility test truck. They wanted it quiet enough to be able to operate it in a quiet subdivision eoad without anyone hearing it or objecting to it.

I installed an ‘extra quiet’ muffler in place of the standard one. The result was, we could run the generator and carry on a normal conversation while standing right next to the generator. 10 feet away you could not hear it above a whisper.

Start with a realistic power budget, then select an inverter to suit. Once you know that, its easy to figure out what size battery and charger is required and that in turn determines how small a generator you need. My guess is you can get sufficient power from 2 kilowatt genset. I have also designed DC gensets that can stall a 10 HP diesel when running at full speed before a charge load is connected. The secret is to plan the system not evolve it.