BYOB to Babbo?
I think if you called and spoke to the wine director and explained the occassion you might stand a chance; as long as the wine is not stocked in house.
If you are looking for a reasonable corking fee with great food. Eleven Madison Park charged me $25 to uncork a Vintage bottle of Krug that I was saving for a special occasion and was well worth the expense.
An old posting from 2002 on eGullet would suggest no, you cannot bring your own wine to Babbo.
CH posting on 2005 indicated that it was still in place: