We were confusing our French terms. Of course they look relatively different, but they are phonetically pronounced coo-de-gra and coo-de-ta, respectively, thus sounding very similar. Their meanings are very different.

coup de grâce is a “blow of mercy” or a mercy-kill.

coup d’état is the sudden overthrow of a government.

In the situation that made me look this up, there was a coup d’état of a group… I’m surprised the new leadership was not also the coup de grâce.