If you think about it there should be a very short list of people who are trusted enough to be shooting skeet with any US president. After all, during the shooting you've got some visitor with a shotgun within a few feet of the President. Common sense and basic security training suggests that whoever is in that situation must be 100% trusted - not only for intent, but for plain old safety. So Dick Cheney, for example, wouldn't make the cut.
I'd like to see the list of people who have been shooting with the President, if there are more than a few people, except perhaps for members of his security detail I'd be surprised.