The furnance fan and electronics are all 12VDC. If your battery is big enough, you might get one night of heating from the furnace before you'd need to charge up the battery. You could do that during the day.
As for the size of generator, if the dollar difference is not that much, then I'd opt for the 2KW unit, even though there is a bit of a weight penalty with the 2KW.
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Cheers
Helen & George VE3INB and Max (Bichon Frise)
2006 Silverado 2500HD D/A,
2006 Cruiser CF30SK
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