Yeah, brasso with 0000 steel whool is the thing! Take pictures before you take the bridge off. I have also (Egads!) used Tarnex on the old style logos that were silver, but don't know if that would work on the newer ones.
That will look like new when you are done!
Sometimes the bridge pins are hard to get out. I have several tiny punches I use. The pins go in one way. Look for the tiny little tool marks on the pins and call that the bottom and tap them out at the other end while supporting the bridge at the bottom so as to not block the pin you are working on, if memory serves me right.