The real question for me is whether you have any completely blown out highlights on the eagle's head. The image looks good, but on my monitor I can't see any image detail in the sunny white area on the head. That indicates that portion is blown out highlights and needs to be fixed. Blown highlights on the subject are never okay.
My goal when making the image is to have a small amount of blinking highlights with the Standard Picture Control on my Nikon camera. Different cameras have different degrees of recovery possible with slightly blinking highlights. I know from experience that I can recover 1-1.5 stops with no problem with cautious local editing.
When editing, to recover highlights I avoid any global edits that increase contrast. That means no auto, clarity, contrast, texture, dehaze, whites, or blacks. If I have a problem image, I switch to a Neutral or even Flat Profile in LR. These profiles give me a starting point with less contrast. All of my edits are done in a way that recovers or preserves the brightest (and darkest) areas.
My edits when I am trying to preserve highlights place an emphasis on local edits - sometimes with a brush at 200-300% to refine a selection.
If all of these steps are not enough, I still have choices. I can clone in color for some reflections with a low opacity, or I can clone in detail from another area that is not blown out. Cloning detail is usually applied with low opacity - I just don't want to see blank areas.
I've spent a lot of time with Arthur Morris going through images and on post processing. From his standpoint, he tries very hard to have a small amount of zebras (blown highlights) on his Sony A1. He has live zebras, so it is easier to get a correct exposure. Once he has a correct exposure he is either using full manual exposure or specific settings with a floating ISO and exposure comp. If he has a blown out area that can't be fixed, it's a discard.