This is an interesting way to look at things. Twilight does fit the pattern of the Beauty and the Beast myth in a sense. If you think about it, Edward, for all his good looks, is still a vampire. In other words, he is a monster by nature. To be sure, he tries his best to not drink human blood, but what he is is not in question.
With this stated, Bella still loves him. This makes Bella the Belle (French) figure in the Beauty and the Beast myth. The names are even similar and her name even means beautiful in Latin as well as the languages that are derived from Latin. In addition, Bella is able to see through the external of Edward and she is able to see his inner beauty, which is the embodiment of the Beauty and the Beast mythology. In fact, this theme can be seen in various ways in the saga, if we include the werewolf, Jacob.
Well if so then the Beauty would be Bella and the Beast would be Edward. He could not help himself but to be drawn to her. Edward seems to be a dangerous fellow but just like Edward can't help but be drawn to Bella, the same is happening to her. Bella is fine with throwing away her life just to be with Edward and once Bella accepts Edward, he in turn becomes a warmer person.
The Beast would represent Edward. Although Edwards is visually not the Beast his personality has all the characteristics of the Beast. Edward keeps his distance from society mainly because of what he is, a vampire, just like the Beast. Before Belle the Beast is usually cold and reserved just like Edward.
The Beauty would represent Bella. Bella had the courage to approach Edward and in doing so accepted him for what he was just like Belle accept the Beast for being a Beast. Also just like Belle, Bella was willing to give up her life to be with the Beast.
With Bella's acceptance the Edward becomes more of a caring person. Although the seemed dangerous both Edward and Beast in reality are actually caring, kind, and soft. Which was brought out with the help of Belle and Bella.