This is just simple math...and having a decent tool to help with Ohm's law.
Start here:
Your coffee pot has a power consumption number in watts. It's printed on a "data plate" somewhere on the coffee maker. For argument's sake, let's say it's 1000 watts.
Plug 1000 watts and 12 volts into this calculator (you are ultimately converting 12 volts to 120 volts, so it's the 12-volt number that counts.)
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I get about 84 amps.
And you know that's about right, because your 2KW inverter is connected to the battery with 00 AWG or similar sized cable...it's huge wire to handle the current.
Your battery is rated in amp-hours. How many amps can it deliver for an hour? And/or how many hours can it deliver a lower number of amps?
Let's assume your battery bank (2 x 6-volt batteries) is rated for 150 amp-hours. Your batteries should have a label stating their capacities. A basic, 12-volt group 24 delivers between about 70 and 80 amp-hours, so 2 x 6-volts can probably double that...but your batteries will tell you for sure.
As a crude rule of thumb, your batteries could reasonably deliver about half their rated amp-hours before being technically "dead" without damaging them. In a 12-volt battery, fully discharged is 10.5 volts. Anything more can damage the battery.
With these assumptions, you have approximately 75 amp-hours to work with. As you can see, your coffee pot will drain your batteries if it's on for about one hour.
The coffee maker needs 84 amps, but for less than an hour. Let's say that the coffee makers needs 10 minutes to brew a pot at full rated power. 10 minutes out of an hour is 1/6 hour x 84 = 14 amp-hours. Obviously, if the coffee maker needs 12 minutes instead of 10, that's a big difference in very precious "amp-minutes". Clean it with vinegar.
Now lets say that you leave the coffee maker on for another 10 minutes to keep the pot warm. Let's also assume that the warming plate needs just 500 watts, or 42 amps at 12 volts.
(Let's also assume the water heater portion of the pot shuts off once the water has run through it...perhaps a big assumption for a cheap appliance meant to hook to an "infinite" power supply--the grid.) That's at least 7 more amp hours--possibly 14. You see where this is going.
Altogether, you've used 21 (or 28) amp-hours out of a potential 75. That pot of coffee is pretty expensive in amp-hours.
Now, let's assume you drink coffee ONLY in the morning so the solar panels have all day to top off the batteries for things like the furnace, hot water heater igniter, fridge igniter, and water pump...and lights, TV, radio, etc. Under decent solar gain conditions, your daytime use will be replenished as you consume it. Unless you're a profligate power user during the day, your 350-380 watt solar array should not only sustain all daytime use but also produce enough surplus to fully charge your batteries. You should hit a point about 2 or 3 hours before local sundown (solar panels shadowed by trees, etc.) where your batteries are full. You may have a solar charge controller that reports the status of your batteries (mine does).
You consume (deplete) 12 volt power all evening and night (when the sun's down) - to run appliances and lights and arrive at morning, desperately needing coffee, a toilet flush or two, some ignitions of the hot water heater, and pump draws (at least one to fill the pot!!
) - with what's left. If you MUST watch TV all evening, keep all the lights on, and so on, you'll need to do the math on all those draws before you know what you'll have left in the battery bank for the morning. For example, in a PUP, the furnace draws about 2.5 amp hours for the fan and igniter. With a 50% duty cycle overnight, that might come to 10 amp-hours or so. A central, ducted furnace will consume more, because the fan that pushes air throughout the living space needs three times more power.
Will you have 21 amp-hours left in your battery bank? Very likely you will if you are careful with power through the previous evening. But if you aren't careful, you may not have even the 14 amp-hours needed to brew the pot.
Plug in
real numbers and do your math...on all your draws.
In the meantime, if you don't have one already, trade in your current coffee pot for one with an insulated carafe. Then you can cut 7 to 14 amp-hours off your consumption by not running a warming plate. You need only brew, then turn off the inverter.
You mentioned a generator, so it's not as if you'll go without your coffee, but over-discharging your batteries can be hard on them, and if the solar can't fully recharge them to "saturation," you might get sulfation...a common problem for solar charged battery banks that never see "shore power." If you deplete the batteries, I'd bite the bullet and run the generator to recharge them for 3 or 4 hours and let the solar top them off.
Or, so you can have it all on power use, you can add two more batteries.
Bear in mind, however, that adding two new batteries to supplement two "old" batteries is not recommended. Ideally, you'd install 4 identical new batteries all at the same time. If you just add two, the old ones will "drag down" the new ones somewhat. Ask a battery expert which of the two should be in series and then which in parallel to get the best performance.
In the meantime, have you considered French Press? We make espresso/lattes in our PUP using the generator, then we make our second cup with the French Press. It's a little chewy at the bottom, but it's really good coffee!
We have a 2 KW generator, and I hate the noise. We only run it to make coffee! When we boondock, I use a 100' 12-3 lead cord (good for the 15 amps +/- the generator delivers) to move the generator away from the PUP. We lock it to a tree with a 1/4" case hardened chain and locks. The inverse square law says that if you double the distance the sound energy (SPL in dB at the rated distance) is reduced by a factor of 4. Physics works. Distance substantially reduces the noise--and you might be able to use an obstruction (hill or embankment) to further reduce the noise. ...as long as you don't have close neighbors.
where:
is the rated discharge time (in hours),
is the rated capacity at that discharge rate (in ampere hours), 
is the actual discharge current (in amperes), 
is the Peukert constant (dimensionless), 
is the actual time to discharge the battery (in hours).
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