Hi John,
Thanks for the post, and challenge!
I think your instincts are right on the money in terms of cropping differently from the original shot, and your other adjustments help bring out the best of this image. The crop actually looks pretty good to me. The setting and lighting are definitely stacked against you, but many times, you just do what you can with the circumstances you're facing.
You've identified one of the problems here - the fence screen pattern in the background. The other thing that caught me eye was the dark, shadowed appearance of the bird's head. Fill-flash would definitely help in a shot like this, so keep that in mind for the future.
I copied your image into PSCS2, and played around with it. To deal with the fenced background, I "defocused" it using a program called Focus Magic. I have a trial version of it, and thought this would be a good test case. You can probably get very similar results using Gaussian Blur.
In any case, I created a separate layer with the blur added, and then using the eraser with a small, soft 'brush' (I use a Wacom pen/tablet), I painted in the areas I wanted blurred. As you point out, too much can be a problem. And I used the levels tool to lighten the bird's head and neck, although there wasn't a lot to work with there, and again painted in just that area, leaving the rest of the image untouched.
Here is the result:
It's a quick effort, starting with the downsampled image you posted. With the original file and more time, a better result could be achieved, but hopefully this is enough to give you some ideas on what direction to go, and how to get there.
In re-reading your post, I wasn't sure if you wanted to blur and eliminate the vertical fence posts, rather than the screening. If so, that would be a major challenge; do-able, but with considerable effort, and of questionable value. And it would represent a pretty major alteration of the original scene.
Thoughts?