Animated Children’s Film with Lost Pet?

Okay here’s what I remember:

It was an old, flat animated children’s film that I think? was about a lost pet.
Something about being lost/abandoned by a road side, the animal is forced to take shelter in the woods. The only thing I’m sure of is the musical sequence where it starts to get darker, and there is a song about how the forest is dangerous at night. During the song there is a picture of a rabbit, which starts off cute and slowly morphs into a fierce looking thing with long teeth and claws!
It really scared me as a kid, but that’s all I remember!

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