Since the distinction between "dwarf planets" and "planets" is still very much up in the air and unresolved, this becomes a very difficult question to answer. Any moon that is round, if it orbited the sun independently, would be categorized by some astronomers as a "dwarf planet" if it didn't "clear its orbit" and as a full fledged planet by those who reject this definition.

If the moon orbited the sun independently and was not round, i.e., has not achieved hydrostatic equilibrium, it would be neither a "dwarf planet" nor a "planet" but simply an asteroid.