Well, there are (at least) two aspects to a technical/general definition of a melodrama. One is that emotions are exaggerated in the acting style and the second is that the emotional tone of the film is driven by music (or the music/score is fundamental to influencing the emotional impact of the film).
I truly don't think that BBM falls into the first part of the definition. In fact, I think the emotional aspects of the different characters are usually very understated.
But as far as having the music play a fundamental role in the emotional direction of the film, I think that's pretty accurate in the case of BBM.
Melodrama isn't necessarily a put-down or a negative thing. There are lots of great, classic melodramas. But, it does has some pretty specific aspects as a genre.
In general, I think BBM is much too subtle to really fall into the definition of melodrama.