A CPL would do f-all for an image like this. You guys can keep dancing around it, but I won't (and don't!), this is just a textbook scene where the DR is really really tough for most cameras. Blowing highlights does no favors for post work, and the best thing you can do is simply not blow them in the first place. Adding something that further lowers your available DR is antithetical to your goals.
The moment is indeed nice, but everything else is working against it. If you have no nature-journalistic integrity, you can work way less, knowing you'll just replace 2/3rds of your image later. If you care about capturing something real, you need to set yourself up for success from the get-go.
If you think the advice of "improve your fieldcraft and put the sun behind you" isn't the best advice for a budding birder, you're beyond help.
All of that said, I don't think the original image is beyond saving, I think it just lacks a bit of savviness in post.