The first step was to create a stack by layering the two images in Photoshop. I used the trees as a background and added a few Curves Adjustment Layers to fix tonal range, color cast, and add contrast (you could also do that in Lightroom first.) At the top of the layer stack is my eagle fight shot. I moved the eagles into a possible position and am now ready to try a blend mode switch.
With the eagle layer active I changed the blend mode to Darken. The Darken blend mode works by displaying only the parts of the active layer that are darker than underlying pixels (in lower layers.) I thought the eagles are darker than the sky so maybe it would work. Unfortunately, the sky in the eagle layer is darker than the background sky.
To fix this little issue, I added a Curve Adjustment Layer and set it to affect only the eagle layer. To do this click on the the click to clip to layer icon that looks like a dark circle on top of a light circle at the bottom of the Adjustments panel or hold down the Alt/Opt key and click on the line separating the eagle layer from the Curve layer (you will know you are in the right place when your cursor changes to the dark/light circle icon.) When we set up a clipped layer the adjustment will only effect the layer it is clipped to. In this case, the curve only modifies the eagle. As I drag the curve up notice less of the sky around the eagles is visible.