don't think your pup has an electric heater. nearly all RV's come with only a propane furnace.
the only electric heater would be one you bring from home or a heat strip added to the a/c.
unless you have the Cool Cat but that's the same as running a heat strip. i don't know how much power it takes to run it but i doubt 1000w is enough.
your generator will be fine for recharging the battery but little else.
i always recommend the Honda 2000. double the power to run the microwave or coffeemaker for only a little more than the 1000w.
and Zoe the Wonder Dog(R.I.P.)
'07 Roo 23SS pushing an '07 Chevy Avalanche
Equalizer WDH and Prodigy BC