The little plastic screw takes a delicate touch. If you push too hard on the screwdriver, you are pushing the screw back in as you are unscrewing it. You need to push just hard enough to keep the phillips head engaged, but not too hard that you are pushing the screw back in. The plastic threads are very minimal. The little clip is designed to install by just pushing the center screw into the clip.
If you really cant get it, you can pry/dig the little screw out of the center of the clip by prying up on it with a pick or tiny screwdriver. The screw part will slide up and out of the larger circle part.
It looks like this on the otherside where you cant see:
Mike W