Here's one of the two original eyes at 100% resolution:
Now, to that original eye, I used Photoshop's Red Eye Pupil Quick Select & Fix tool with the parameters Pupil Size 70% and Darken Amount 80%. I selected the entire pupil and a bit around the pupil when using the tool.
In this last instance, I took the original, made a selection
slightly larger than the pupil and copied it to its own layer twice. So now we have original image of original eye on the bottom layer, and Layer1 of the pupil (pupil and a bit of iris) and an identical Layer2 of pupil.
To
red channel of Layer2, I applied the green channel of Layer1.
To get this following result:
My channel method takes only seconds.
This photo was taken with a point and shoot with the flash on. The catchlight/flash burn remains intact, as it should. It will be 255 in the red and green channels.
You can judge which result you prefer. Perhaps using different settings would yield a better result with the Photoshop tool. Perhaps. But I am happy with the Channel method.