Looking at this from the outside, it seems to me no matter what this kid needs to have some sort of very big consequence for what he did (assuming you are certain he did it). That is such a bright line to cross that I have a hard time understanding why any leniency would be shown, but I realize I cannot fully put myself in your shoes, with him being like a second son.
So I ask myself what I would do if it were my own son who forged my signature and a letter, and I think I would probably turn him in to Clemson, even though he was my own child, and even though I would be very upset. I am a strong believer in consequences for ones actions (positive and negative). Getting away with fraud and dishonesty is a terrible lesson to learn. Better to learn the other lesson – that you don’t get away with it. Much better to learn this now than 10 years from now when the consequence could well be jail time.
ETA: the boy not only betrayed Clemson, equally importantly, he betrayed YOUR trust! I think I would bar him from your house going forward too. I am all for helping people who need it, I have taken in many kids needing a place to stay (under different circumstances) and I would never let someone back into my house, into my trust, if they did something like this. Life lesson: setting good boundaries for yourself is important!