The OP has already discovered the burglar so he can't use his cat like instincts to surprise him. The burglar is in the house committing a crime. Anything the burglar says can be assumed to be a lie. Who would trust a burglar? I would not want to let him just walk away. When the cops come, they just take him to jail where he probably gets better treatment than on the streets.
I discover a burglar in my house who says, "Don't shoot me, I'm only going to rob you." This would mean that I'd have to stand there and watch myself being robbed and as long as he didn't try to inflict "great bodily harm" on me or my wife I would not be justified in shooting him. He could stuff his bag until he heard the sirens coming.