A quick rule of thumb for working out input on 12 volts is output watts/10=input amps. not 100% accurate but will get you into the ballpark for working out cable runs.
For something that big I wouldn't bother running a fuse but go overboard with the cable insulation, a 3kw inverter would have a 5 to 6 kw surge rating, your talking more power than the starter motor, several times more and is there a fuse in the starter cable......no there isn't.
Also running your engine while you use it, you will stress the bcdc running it at max output but theoretically it should cope, but it will amount to a spit in a bucket as far as the draw required and if you bypass the bcdc while it is running you will need to keep the engine revs well up so the cooling fans inside the alternator will move enough air to try to cool itself, alternators can not run at max output for very long.