Champagne is just wine that has been carbonated by fermenting the wine a second time. If you choose to include wine in your Paleo lifestyle, you can certainly include the occasional glass of champagne.
The Health Benefits
In addition to acting as a probiotic, wine comes loaded with antioxidants that have been shown to protect against aging and Alzheimer’s (source).
Red vs. White Wine
The down side to champagne, is that it is always made from white wine. Red wine is preferable because it is lower on the glycemic index and has more antioxidants than white wine. That doesn’t mean you can’t have white wine or champagne, just that you should limit them to special occasions.