Wow. Really nice piece. Please don't polish it. It didn't get that color overnight and once messed with, it will take many years to even get back to where it is now, if ever. I'm a plaque nut and have been for over 30 years. Blue Ribbon is the only thing I would try and even with it there should be no rubbing of any type. Put a few drops on the problem areas and roll a Qtip over those areas. Let it set a few days and do it again. Never rub, just let the rolling Qtip pick up the loose garbage. It's not a fast deal but it works. The problem areas will never be the same color as the rest of the plaque but that's better than a shinny piece of stuff. As a collector, I would buy it as is but not after being messed with.
Probably the best advise so far. If I were you I'd find out if that thing has a collectors value prior to any experimentation.