I love watches. My collection: Breitling Colt, Bulova Accutron, and a few others.
Yours is no slouch either. I believe it's around $2500. and the band is special, very special. It is made from ceramic and can not be replaced with any other band except for the one it came with. That being the case, if you are going to pay that much for your watch you should expect some quality service from the maker in regards to things like this. So before you do anything else give them a call and ask them if they have any recommendations about cleaning it.

If you get a response some thing like "No problem just send us a blank check and the watch and we will fix it right up, Honest" tell them to get ben........never mind. Just hang up. Their # 1-888-455-9200
If you are going to try and clean it you need to understand just how hard ceramic is. I have ceramic tool cutters almost as hard as diamond cutters so it will take some patients.
The first thing I would try is to let the band soak in bleach. You might be surprised at how well that works. Let it soak awhile. The longer the better

The best thing is to find a polish made specifically for ceramic.
I would try a auto paint supply store. They should have good quality polish for ceramic clear coats. It may also be a good idea to get a piece of white ceramic cutting plate/board and test a spot. I would also use a small Dremel polishing wheel. Dremel also makes a soft polishing stone that you mith want to try depending on how bad the stains are.

Remember a ceramic cutter board is a cheap way to test before you try anything on the band.
I really hope something here helps.
Let me know if you need any other help and please let me know of any success or failure and I'll try and give additional assistance.