Depends not just on how good your accent and pronunciation is, but also what the song is, what the mood of the song is, etc. For instance, I speak pretty good French and my accent is pretty good, so I might perhaps risk singing a comic song or a simple love song at a French singaround, if I ever found myself at one. But I think if I were to try to sing something tragic or overly serious I imagine I would probably sound foolish at best... or slightly grotesque at worst.

You just have to imagine, in your head, a song you deeply love that is very poignant or moving or chilling or disturbing or in any way emotionally intense, being sung in heavily accented English; then you'll have your answer.