You are not quite understanding gravity - all objects with mass have gravity between them proportional to their mass. A flat earth would still exhibit gravity, because it would still have mass. So your bullet would still arc toward the ground, just as is the case on the ball-earth. The ball earth spins however so the landing site of the bullet would appear to be to the left or right of the spot the gunner expects when the shot is fired, this is the coriolis effect you mentioned - it is caused by the spin, not gravity. Of course the bullet travels in a straight line as viewed from above in isolation, but this would look like a curve if you plotted the path on the surface of the earth, because the earth is spinning.

A flar earth could spin also of course, but the earth is not flat, because if it were the same stars would be visible from Australia and Britain, it would also be night/day time at all locations on the earth simultaneously, which it never is)