It’s hard when a chick - your own or one you have taken under your wing- let’s you down like this. I think some of the harsher posts on this thread miss that part.
Coulda/woulda/shoulda: you let this chick into your house and your heart in the hope that a safe harbor would make a difference. He is still the same chick, still has the same challenges. If (per @brantly) you could get him into a room with you & a therapist maaayyyybe some growth and learning could happen.
But if there was every a time to model how to handle ethical dilemmas for you, the friend and your son, it is now. Clearly articulate why it is/was necessary for you to call Clemson to find out if they have received a letter from you. If they have, clearly explain why it was necessary for you to inform them that you did not write that letter- starting with the implications for you.
Because for all the good reasons for honesty in this scenario, the friend has also put you at risk- not just for all the ‘should you use your work computer/letterhead access’ pieces noted above, but because you have a professional reputation that is jeopardized by that letter. There are any number of ways that it could come back to bite you later. For example, somebody on the adcom remembers reading about the supposed internship and tries to refer a student to it- only to find that it doesn’t exist- and assumes that you made it up to help the student. Or just mentions it in passing to a mutual colleague, who knows there is no such program. And so on,