A nice image. Agree re BG.
Darkening works. You may also create a new layer and paint over the bright areas with the adjacent colour, a very effective way of getting rid of bright areas behind the image. If you are worried about painting over the bird, then just change to eraser tool and paint over the bird. Ther are more ways to do this but this is a good place to start . What painting over the back ground does also is it blurs the BG somewhat.