I am jealous....wow....such a close encounter with a Cooper's, fantastic shots.
I think the blur on the BG will work but in this case you may need to do a couple other things.......
1. The bird needs to be a little brighter overall compared to the BG
2. Using more noise reduction and reducing contrasr just on the BG will help too.
Really unusual opportunity ... he must be hunting your song birds, I see these swoop through the tress behind my house and they perch at a distance but never have been this close...kudos for being ready.![]()