In this case, "Lord of Kozuke" is a title (a kind of privilege, a peerage, if you will), and doesn't necessarily mean the smith was actually living or working in that location at that time. In this case the smith has written on the opposite side "Resident of Osafune, Bishū Province", so I think its safe to assume that this was forged in Osafune (assuming always the signature isn't a fake).
My preference / recommendation would be to restore in Edo period fittings. It is a hand-forged sword from Japan's feudal, past. That it spent 5-10 years of its ~300 year life in military mounts doesn't impress me. If the mounts were in pristine condition it would be one thing. But your mounts are in an advanced state of distress, and I think they are a write-off from a collector's viewpoint.
Also, finding a set of military furnishings that will fit this hand-made sword would be a frustrating and, I think, pointless search. The sword was not made as a WW2 sword, so spending money and time bringing it back to an anachronistic condition is not something I would do. If it were a factory-made WW2 sword, I don't think it would be too hard to find replacement fittings. The fittings would be appropriate to the time and purpose and history of the sword. However, your sword was hand-made, so the length and curvature are different from a mass-produced sword.
I think you do your father no disservice to return this sword to its pre-war condition.
But first, you should get your sword looked at by someone who knows their stuff. Maybe there is a dealer or restorer nearby where you live who can help you out. Your sword could be a forgery, as is common with Japanese swords. It definitely needs the cosmoline removed and replaced with a lighter oil, but not until you have a clean scabbard in which to store the sword. If you are lucky you can find a reputable dealer who can take care of all of this at once for you: appraisal, advice, and sending the sword off to a scabbard-maker so you have a clean, plain scabbard made for it. He should be able to tell you if its worth sending the sword off for professional polish. Do not attempt this yourself. A centuries-old sword needs specialized polishing skills, and you can easily ruin your sword by trying to scrape off rust. Avoid the temptation to remove any rust. I would say leave it in the cosmoline for now, since it has probably minimized and stabilized any rust, and since it is probably also coating the inside of the scabbard.
1) Show it to somebody who knows swords - a reputable dealer or collector. The forum can help you out.
2). If authentic, get a plain wooden scabbard (shirasaya) made for it.
3). Once you have it cleaned and stored in a shirasaya, you can decide how much you want to spend on restoring it: polish and furnishings. (The shirasaya also makes it easy to ship, should you decide you want to ship it somewhere to be restored.