I'd fill it with water and really hit it with a plunger.

Look horizontally UNDER the top plunger piece down into the drain hole. the stem that holds it to the shaft that's screwed into the tub drain fitting holds the key. There will be a flat head screw in there (below the black rubber gasket) to undo that will let the plunger piece be removed. The drain itself is likely a Moen part number T90331(then two letters for the finish color, likely AZ antique bronze based on the photo).

Moen's website doesn't show squat for how to do this so you might have to go to a plumbing supply store and look at an uninstalled new one since I'm admittedly doing a poor job of explaining it!