Mystery handle screw on American Standard kitchen faucet

I'm trying to remove the handle on our kitchen faucet. It's an American Standard, single handle, where the faucet rotates independently of the handle (you can put the water on full cold while you have the water coming into the left sink). I can't find the actual model on the American Standard website.
Anyway, to remove the handle, I pulled out the little blue and red button from under the handle. When looking inside the little hole I see the "screw head" but it is a circle head, not anything you can put a screwdriver into. No slot, no plus. I thought maybe it was a hex shape, but in fact it's not. It's a head of some sort with a completely circular indentation.
I am lost! How do I remove this screw, or, how do I remove the faucet handle?
Thanks for any guidance,
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