A trick I just learned is to add the bitters to the sugar in a spoon held over the flute. This avoids the unsightly splashes of bitters down the side of the glass. It does slow the process down if you're serving a number of guests at once. If you use a coupe, you can also avoid these splashes, but I mostly make this drink at a party and my coupes are in use for other drinks.
Some recipes include a splash of cognac before you add the champagne - say 1/2 oz