This is very nice! Looks like you're getting sharper images at 500mm than at 600,consistent with what I've heard about the lens. And a nice BG.
Interesting coloration around the eye -- not like the Mourning Doves I know. Everything seems to have better color in Texas!
I think you could have an option to crop off the right 1/3 to get rid of the branch in the branch (as it were). Not a big deal but the pose here would work with a squarish crop.
I love this image; the detail, colors, perch and the green velvety background. What a beautiful image. Not a big deal at all, but I find the branch and lichen in the upper left corner to be competing with the subject. I might try removing that if possible. I don't know how though, perhaps Diane could comment.
Photoshop to the rescue. Select the Patch Tool (hover over the Band-Aid in the Toolbar and several choices will pop up). Draw an outline a little outside the area and drag the outline down to a clear area. Never mind that it's lighter. Let go and the area will be filled in with a good attempt at matching the gradients. But there will usually be some posterization-type artifacts. I've left them here to show you -- they can be cleaned up with some careful cloning. Increase the contrast greatly with a temporary adjustment layer to see them better while you clone them out.
I also went back to the original and cropped just a little less tight on the right. Then I lightened the shadows a little -- had to use Shadows-Highlights on the JPEG but it's better done in the raw conversion.