I'm not an expert but I would assume the eagle is something like a pewter metal which is cast. Drilling holes in it and trying to reshape it with a dent puller method would utterly destroy it!
If anything I think a tin solder would probably be the best bet to repair it. A couple of months ago I had my eye on a garde reiter lion on ebay that had a hole drilled through the nose and mouth. If I won it I planned on trying tin solder and then silver plating it with electrolysis. I only bid 400$ and it sold for $3200, so I didn't get to try..lol
They have tin solder with different alloy content too, so maybe one with some silver in it could match close to the original....imo ?
Ultimately in the end you might have to grind and sculpt the soldered area with a fine grinder or file to match any feathers, fur, etc.
If it is REALLY dented in and crushed, then perhaps heating the seams and disassembling the eagle would be the next option. Each piece would have to be then gently heated and tapped back into place with wood blocks and jigs but you run a risk of cracking parts. Then the entire eagle would have to be soldered back together.
I would only attempt repairing it if it was a total wreck and your willing to accept a loss if it doesn't work out!
Show us some pics of your eagle !!!!!!!!!!!!!!!!!