You could try using curves on the original but it needs very small changes. Levels are probably easier. Hope you don't mind but it's easier to show you. These were the levels setting.
What I have done is move the bright end of the output down to dim the white and then moved the bright end of the input down until the white starts to loose it's shape.

Turns out that I should have left the black input where it was but the output black level is usually a good idea. I then moved the centre input slider to taste. That gave this
Then thought I had better look at your result. I had dimmed the white down but looked like you wanted more contrast in the dog so used curves. The 3 points on the line split it into 1/4's.
You will see what these points do by moving them around. The white of the dog's eye are within the 3 points. The slope is steeper between them so contrast is increased. Also by moving the black end along a bit. That gave this result.
This may give you some idea of what you can do with a shot like this with these 2 controls that most packages have.
John
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