Yes. It doesn't say 'for the purposes of Smite Evil damage' it simply says 'for the purpose of Smite Evil'. Since the number of uses of Smite Evil is dependent of class level, and the bracelet modifies your class level for Smite evil, you could get 1 or 2 extra uses out of Smite Evil.
Everything that is based on level is increased. If a level 10 paladin has this then he counts as a level 14paladin and would thus have 5/day instead of 4/day.