Adding an extra #10 wire in parallel with a #4 will help, but to be honest it will make little difference. You would need something comparable with another #4 to make much difference.
If you have the #10 around free of charge then use it, but I wouldn't spend money on it for the benefit you will see. I don't know how bad your situation is, you don,t give enough information and I don't have those tables of wire loss handy like some people who are doing this sort of thing every day.
Battery fuse needs to be bigger than the full load rating of the inverter. 300A is probably reasonable, most 12v systems are for smaller installations. 300A should just about handle the surge of a 2kW inverter or even a 3kW inverter that is not overloaded with high starting loads. If you have a much smaller inverter than you could use smaller and have better protection for your cables and battery. Things need to be sized for the individual case.
Flux