I would not try to make the background uniform. That's just a difficult road to go down if that's always your goal, given how nature is not going to play ball with you that way.
Instead, I would mask the background completely and slightly reduce luminance and saturation (it's already soft so no need to do negative texture / sharpness), while at the same time slightly raising those same values for the subject. Making the bird POP with a difference in perceived light and color will do wonders to keep the eye from noticing what's behind, especially since there are no actual spectral highlights. Making those adjustments in small increments will keep it real.
Chris