It doesn't appear to be symbolic at all; a dog barking at a crow is merely an annoying sound that she would rather listen to.
Crows are notorious for teasing other animals. I mean... look at this!
In the above linked video, the crow is hanging around behind a dog, who's tied up, and repeatedly going and pecking the dog with it's beak and dashing away. It just keeps bothering the dog. Presumably the dog is barking at the crow, but the crow doesn't leave.
Beatrice is taking an example of an annoying sound, and saying that she would rather listen to that then the man's declaration of love. She's giving a very clear message - _I don't love you_.
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Here's a video of a dog barking at a crow. The crow is just staying there, while the dog continues to bark.