If it doesn’t pop out easily, as if the clear has sealed it in place, I’d suggest scoring or cutting through the clear along the star plug with a BRAND NEW, SUPER SHARP Exacto #11 blade. Slowly and patiently, like you have all the time in the world. Then pop it out from underneath.

That’s how they come when you buy them new. You just pop the centers out when you’re ready. It’s possible that yours wasn’t cut all the way through? Now, if it’s been cleared over that would definitely make it harder to remove, thus the need for the sharp exacto. I really do believe this is the case looking at your pics.
However, if it really is still molded together and not completely pre-cut from the factory (very unusual) then I’d pop that one out entirely and set in a new one. It’s a pretty easy job, though you should still use the sharp #11 Exacto to cut around the round insert to slice through the clear, otherwise it would lift the clear on the playfield when you tap it out. Polish up the new one so it looks nice and shiny, and install it level to the clear coated playfield.
When I clear coat playfields I use an inverted white star insert because they fit loosely in the hole and have a little handle I use pull it out before the clear flashes. I can see how using the original, tight fitting plug can make it harder to remove.
If needed, once the center insert is out, I will go back with the #11 Exacto and carefully clean up any edges so the new star insert works smoothly. Sometimes I use a small file. Sometimes I use a hot exacto blade (attached to a solder iron type device) It’s just part of the process)