Yes, bring your wine. It is not "only if it is not on the list", but bring something special. Remenber to bring something that cost more than $100 and if it is an old vintage they will probably not have it on the list. I think they will allow only two bottles for the table, so buy a white or champagne from the list and use your two reds. That will cost you about $200.
So..... bring them and Enjoy!!
Food for them is very very serious, so relax, enjoy the wine in a quiet ambiance, and enjoy every byte of the food!!!
My final suggestion is to have the grand chef tasting menu which is about 8-10 servings including dessert and cost $140 but because it is the first time, it is your birthday, and because it is extraordinary food go for it. Also if you like foie gras, then order it for an extra $20-$25. I have been about 6-7 times and almost always order it. We were there last time Oct 1 and Oct 10.