I went to Ace yesterday and found a part that fits perfectly but they were hard to find. Look in in the isle with all of the loose screws. There are a bunch of pull-out drawers with labels. There should be one called "License Plate Fasteners". Hopefully in your store you will see part "No 59700, Insert Nut for No. 8 Self-Tapping Screw Rear: Chrysler". See picture below which shows the drawer label and the bin label too.
I paired these inserts with a #10 5/8 screw from the same isle. These screws worked fine in the new insert as well as the originals without any trouble. Now that I know about this issue though, I did press down firmly on the plate and frame while screwing in the screws so that there was some friction to prevent the inserts from turning.
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