There IS a connection between eye health and the liver, but not so much with the bags under eyes and not sure about the lids. It's more with the health of the eye itself.
But certainly increased circulation, cleaner/purer blood OR conversely, very toxic blood, will change the color of our inner eyelids and even the sclera. I can always tell when I eat something my body doesn't like, as the sclera (white part of eye) will turn red and inflamed underneath the eyelid. Thank gawd it doesn't extend to above the eyelid LOL! :)
I tend to be anemic also, it's always been a tough demon to slay and now with impaired kidneys even worse, so I've been checking my eyelids just like you for iron deficiency. I've yet to notice much of a change either way, so I'm not sure how accurate a tell it is. I think the color has more to do with circulation than actual ferriten levels.