I agree that they're great images. If only the wild ones would cooperate so nicely!
Re the backgrounds: If you're a Lightroom Classic user, Adobe have made it pretty easy to select the subject or background via the relatively new masking tools in the Develop module. I'd suggest creating a background mask and then darkening the background so the raptors pop more.
I do that with flowers, birds and so on, and the auto selections make it very easy. They can be adjusted with brushes if needed.