Yes there is a traditional song called The White Hare. It's in Frank Kidson's Traditional Tunes 1891. It's not what the Lakeperson does. There was a thread only a few weeks ago in which several people (including me, I think) posted versions of the Cornish legend about some drippy woman turning into a rabbit and stalking a bloke who'd dumped her. The vid is sad (derogatory sense) or hilarious depending on what mood you're in when watching. It's a bland, trite pop song of formulaic structure based on a silly story.

Rabbits On A Beach