That is a big inverter.
A 1000W inverter is capable of drawing over 80 Ah out of the battery. If the inverter is 6 feet away from the battery, you would need 4awg wire to handle those 80Ah, per the Blue Sea wire size calculator linked below.
http://circuitwizard.bluesea.com Note that you need to input the combined total length of both the negative and positive wires into the calculator.
Both the positive and negative wires from the battery would need to be sized to the inverter. Also, the shunt would need to be sized to draw that much current, because that much current would be flowing through the shunt.
The fuse for the battery would need to be sized appropriately at 125% of the expected maximum current draw.
An electric coffee pot could easily draw 80 DC Ah.
If you wanted to power a device with an AC motor, like a small 120V refrigerator, the current draw could be three times the rated Ah.
In any event, your inverter current draws will determine the size of the wires feeding the inverter, the size of the shunt and the size of the fuse at the battery.
As others rightly pointed out, all the other wires would need smaller fuses matched to the applicable wire size, with each fuse sized appropriately at 125% of the expected maximum current draw.