…or do I have to choose a face-down Keeper, and hope I get the one I’m looking for?

A:You have to select a Keeper, not knowing what it is, hoping you remember its placement. Part of the point of Government Cover-Up is to make Steals and Exchanges more difficult for one’s opponent.

A player could even deliberately mix their Keepers around after turning them face down, if they wanted to (then check their locations, since one can look at one’s own Keepers any time). Remember, everything that’s an advantage for your opponent under Government Cover-Up, is also an advantage for you if your opponent were in a similar position to Steal one of your Keepers.