FWIW, have you ever seen a barber use a compound on the leather strops that they use for their razors? i never had. You CAN use a "conditioner" like Lexol or other leather conditioners to maintain the suppleness of the leather but a stropping compound is entirely different.
A stropping compound is an abrasive and there's no need to use one if all you want to do is restore the sharpness of an undamaged edge. On the other hand, there's no problem using a light abrasive compound to enhance the stropping (which is less damaging than using a stone) but once you use a compound on a strop, the leather isn't good for anything else. In fact, if you use different compounds, you need to a separate strops to use for each compound; not a good idea to mix them.
Can't advise you on what stones to use for your Japanese blades, especially if they are damascus. I just looked and there are a lot of videos and articles about how to sharpen Japanese blades but the "advise" varies widely depending on the type of knife, the grind and the blade material. I'd study as many of these resources as possible before attempting the sharpening of any "valuable" Japanese blades myself.
Fortunately, most of my kitchen knives are just made of cheap carbon steel and they seldom need more than a touch up w/a sharpening rod or strop and I seldom need to resharpen them w/a stone. So, it doesn't much matter that I do to them.