Welcome to BPN, Richard! Great first post, I like the details and pose. My first thought was that it needs more contrast. I don't think that increasing it will make the eyes less prominent. Did you desaturate the snow? Keep them coming!
I got a lot of blue on the snow, probably due to the blue sky and I have to admit that I don't know how to remove it without touching the snowy owl at the same time...
I like the colors in the original better. I increased the saturation of the eyes, adjusted the black point, increased contrast and sharpened head and body more.
In your RAW file you can desaturate the blues if necessary, sometimes raising the color temperature a few degrees helps, too.
Welcome to BPN! Terrific subject for your first post. Your white balance looks good on the original post, but I think Axel's repost with more contrast and sharpening is an improvement. If you are using Photoshop for processing, it's easy enough to create a Brightness/ Contrast adjustment layer and then mask out the bird. Look forward to more of your photos!
Welcome to BPN Richard, and what a gorgeous first post! Lovely flight pose and excellent composition. Axel's tweaking in PS made the image even better. Looking forward to more of your images!