In response to you asking, are you really being a parent if you don't let him come home:
Remember, you are being a parent to your two younger kids. A very good parent. You are keeping them safe.
My opinion (remember, on this board, take what works for you and leave the rest) is to let him stew for awhile. I would not beg him, not one little bit. I would not buy him shoes or dinner. I hear your reason for buying the shoes. OK, now he has them. Don't buy him anything else. I would have let him run track in whatever shoes he already had. Dinner? No way. Go somewhere and get a pop (or soda, or coke). This is to send him a message, not to be cruel to him.
If he asks to come home again, I would tell him, certainly. I would like you to sign this. And I would have something ready for him to agree to. And I would not entertain his return to the house until and unless he agreed to it.
Him returning to the house is equivalent to his returning to the family. If he cannot or will not function as a member of the family, then he cannot stay in the house.
Again, not to be mean to him. But to let him know that his actions effect the entire household. And the family is not going to tolerate it anymore.
Hugs and prayers.