Based on my research, my advice would be don’t go too dry (as in, don’t choose an Extra Brut champagne). Beyond this general advice, Blanc de Blanc champagne seems to be recommended most when it comes to pairing champagne and lobster. Chardonnay generally goes very well with lobster and, given that Blanc de Blanc is made from 100% Chardonnay grapes, it is a great compliment to crustaceans.
Of course the type of lobster dish being served should also inform which champagne you choose. According to champagnedirect, Blanc de Blanc tends to go well with cream and spice-perfumed dishes and would work well with all three of my favorite lobster recipes for New Year’s Eve. Rose also goes very well with lobster. I’d pair Rose with lobster for a lobster bake (also known as a clam bake) or summer BBQ. As Non Vintage champagnes go well with nutty dishes and cheeses such as Gruyere, I would serve a Non Vintage champagne with lobster macaroni and cheese or a lobster grilled cheese sandwich.
UPDATE: I’ve recently written a more detailed article on which wines and champagnes go best with specific lobster dishes, including rationals as to why and suggestions on specific wine brands/bottles here and an article with five champagne pairing suggestions from a champagne specialist here.