I'm sorry this happened. Something similar just happened to me an hour ago. You are certainly within your rights to refuse the cancellation.
This is how I evaluated my own situation. Yes, I'm losing $1150 from the booking that just got canceled, but forcing the people who lied about my property to go through with the stay would be so much more stressful. I really don't want people who are hostile to stay there. So I canceled. But you will be losing five times as much, so it's a more difficult decision.
It sounds like you already tried to reach a compromise? All you can do about their complaints to CS is to provide them with proof like the logs of the lock that they put in the wrong code.
If they already left, you can hold onto your guns and refuse the cancelation. At the end there will be a negative review either way. Remember, customer support may overrule you and give them the refund anyway! Hosts are at a disadvantage like that.