Dry camping with residential refrigerator

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Yes, in theory a 1000 watt generator can produce up to about 80 amps to charge your batteries through the converter.

In practice, the converter's efficiency and power factor will limit the charging capacity to about 50 amps or less ... If nothing else is on.

If you have other users turned on, they will reduce the power available to charge the batteries a like amount.

This isn't a concern with a 2000 watt or larger generator, but may be a factor with a 1000 watt generator.  If you exceed the generator's power rating it will either trip a circuit breaker or shut down.  My Honda 1000 required stopping the engine and then restarting to recover from an overload.
 
Your inverter for the fridge may or may not also be the charging system. I suspect not, and you probably have a separate converter/charger and maybe a 55A charge capacity. That means an actual charge rate well under than for most of the time, but maybe a peak of around 40A when you first start up the genset and plug the RV in. That's 40A@ about 14vdc, do maybe drawing 5A-6A from the genset (600-720 watts).

A 2000W genset may not be able to start the a/c unit without tripping its breaker. The compressor start-up amps will briefly exceed 25A and the gensets own breaker may trip before the load settles down to the running amperage (11-12 amps@ 120v). The 2400 (2000 continuous) or 3000 (2800 continuous) is a better choice. Be careful of the genset ratings - many models advertise their peak load capability, not the sustained or continuous load amount. The 2000W Yamaha inverter/gen has a 1600 watt output, close to what the a/c uses continuously (about 1440 watts).
 
Well, I took the plunge and got a Honda 2000 and left my two 12v deep cell batteries as is.

We've used the generator twice so far and it managed to run the refrigerator and some appliances without much problem, though it runs through the gas pretty quickly. I got about two hours on a gallon.

An RV salesmen who sells my model says I should keep the inverter on when I run the generator, presumably to keep it from overloading the generator, since the inverter is only 1000 watts. Does this make sense? The inverter is only for the fridge. I haven't tried running it yet without keeping the inverter turned on.
 
That gas consumption is way out of line for a Honda 2000i. It shouldn't use more than 1/3 gallon per hour when running flat out under it's maximum load - that's 3 hours per gallon.  Under lighter loads with the eco-throttle on (so the engine doesn't run full speed all the time) you should get up to 8 hours per gallon.

Are you turning the choke all the way off once the engine is running?  If so, I'd take it back and let the dealer figure out what's causing it to use so much fuel.

The only reason to run the refrigerator via the inverter is if it's starting surge causes problems to the generator.  The downside is the current the inverter pulls out of the batteries has to be made up by the converter/charger, so it may take longer to charge the batteries.
 
If the inverter is an "In-line" model, that is one that takes power in, charges the batteries, when power is available,  And which switches to INVERTER mode when shore/generator power is lost.. Leave it on and let it auto-switch.

120v--inverter---120v
. . . . . . . 12v . . . . . .

if it's a stand alone (12v---inverter---120v)

Then turn it off

(I hope that top one came out right, then 12v is supposed to be directly under INVERTER.)
 

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