The high polish on the blade should make it simpler without altering the existing finish, using polishing paste like Flitz, Simichrome, etc. The description in the named auction states 'a few spins' on a buffing wheel could clean it up; that might work too, with some polishing compound (white rouge, etc).
It's possible, maybe likely, there'll be some pitting left behind, even after polishing away the dark oxide or rust in the spots. But the polishing should help make the spots (or pits) less conspicuous. Only way to remove the pits is to sand the steel to the depth of the pits, and completely refinish and repolish.
BKF (Bar Keeper's Friend) can work very well to remove rust. But it's also capable of etching steel; you have to be very, very careful with it, so it doesn't alter the high-polish finish in it's own way. An 'etched' finish often looks like bead-blast finishes, kind of a dull matte grey; it's easier to conceal on satin or bead-blast finishes. But on high-polish finishes, any subtle etching will stand out more. Heed the instructions on the container, which recommend not leaving it in contact with the steel for longer that maybe ~ 60 seconds or so, then rinse it thoroughly away.