Hi, I recommend using wd40, spray it on a q-tip or cotton wool and rub it on. I tried it personally on my Floyd rose bridge and it cleared up well. First hand experience.
I also tried brasso which worked but I think brasso might be more damaging, not sure.
Steel wool I have only tried on my frets. Worked good, but for fret I prefer using brasso.
I don't recommend using steel wool, it won't leave scratches if u use the 0000. HOWEVER BE WARNED that when u using steel wool, there will be power like flakes of the steel wool that comes off, and That shit is dam annoying, I got it on my lap and tried to sweep it away. I think it was so fine that it got into my skin, I had rashes after that on that area. I didn't know what to do, I used a vacuum cleaner and vacuum the shit out of my skin LOL, after that I used a super strong magnet an try to remove all the powder flakes. Luckily it worked, I actually saw little silver dots on the magnet.
So I highly do not recommend using steel wool. Especially without proper education on how to use it