In these situations I try to remember that I am their teammate, and part of my job is to make the team work as efficiently as possible. You mistrust your teammate, and obviously he mistrusts you, or he wouldn't try to hide the problem. The trust needs to be repaired both ways, not just yours for him.
My recommendation is to think about what you can do to help him, which helps everyone on the team and ultimately, your clients. I would schedule a postmortem with the team to go over what happened and to work together to find some ways that the whole team can prevent these sorts of errors in the future. You've already mentioned that it makes your team look bad, so really, all four of you are responsible for the problem, not just your teammate. Surely there was something someone on the team could have done that would have caught the problem before a customer did.
When he realizes that you are more focused on making sure the team's work is correct than on blaming him or making him "own up", he may come to you for help instead of hiding his mistakes. Going to your boss just reinforces that you don't respect or trust him, so I would advise against that until you've tried to repair the relationship.
Yes, it's unfair that the team may have to take on some of his work because he's not doing it well, but which situation is better? The one where he pretends nothing is wrong and keeps making mistakes and covering them up or the one where he comes to you for help and starts learning how you and the other folks on the team avoid these sorts of mistakes? Maybe the team could use this opportunity to talk about ways the whole team could work a little more efficiently or effectively.