Sabatical said:
A 1000w continuous inverter should have a 200a fuse or 175a breaker on 2/0 wire.
The size wire you need depends on how long the wire is. If you put the inverter 6 inches away from the battery #6 might be fine and is much easier to work with.
The power companies use high voltage transmission lines. The idea Tesla had was to use AC and transformers to step up the voltage then send the watts with low current over skinny (cheaper) wire. Edison wanted to use DC and put a power plant on each city block. You have low voltage DC which means high loss to transmit the power. Put the inverter at the battery then run the 120 volts at one tenth the current to get the watts to the coffee maker with one tenth the wire size.
Air conditioners, refrigerators, and anything with a motor will have a high starting surge. Computers and microwaves have starting surges. Tungsten incandescent lamps have a surge while warming up. Resistive loads like coffee makers, rice cookers, toaster ovens, have no starting surge. It takes 600 watts to run and it takes 600 watts to start. The inverter, like a computer or microwave, has a starting surge but not the coffee pot.
You can't have electricity without math. #6 wire is .4 milliohms per foot. Two one foot pieces of #6 wire, one for plus, one for minus, is .0008 ohms. At 50 amps (E=IR) the voltage drop is .04 volts. That is negligible. You can use 2/0 wire if you have hurt feelings about crummy #6 but the performance will be the same. Now if you put the inverter 10 feet away from the battery (20 feet round trip) 4/0 wire won't be as good as 1 foot (2 round trip) of #6. Bigger diameter wire helps but cannot make up for length that is unnecessary. Is there a reason the inverter has to be far from the battery? You can have it far but 4/0 wire is almost 1/2 inch diameter, heavy, expensive, and not as good performance.
You want a fuse not a breaker. If the circuit is overloaded and the cost of replacement fuses is too much a breaker will, at high cost, save the pennies that fuses cost. It is better to fix the circuit than to adapt to overloading it. Breakers have many parts that can fail. They can fail open circuit, safe or they can fail closed, fire hazard. Breakers have more voltage drop than fuses. Some people like breakers so they can use them like a switch. Don't do that, repeated tripping of the breaker will get it to failure sooner, on purpose.
My advice is before you spend kilobucks on kilowatts of sine wave inverter or replace battery or add solar try this first. Put the inverter where you can wire it with #6 or #8 wire as short as possible and then try brewing coffee. The plus and minus wires don't need to be the same length. The only reason to make either wire any longer than the absolute shortest possible is to keep it safe. Sharp edges of steel will cut insulation. There may be other hazards. You need to be able to check and add water safely.
If the inverter is fused internally and the wire is run in a manner where it cannot possibly make a short circuit, for example the wire is only 5 inches long and cannot possibly reach any other conductor, then it may be reasonable to not have an additional fuse. The wire needs to be protected from other conductors reaching it. A good example of this is where people wire a pair of golf cart batteries in series with a short unfused jumper between the batteries. It is too short, even if a bolt falls out, to reach anything.