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Make sure you follow up with Barclay, the fact their letter was dated same day you sent out your verification sounds very likely they made their decision without your proof.
Open a new case if necessary, be on hold ti you get a supervisor, etc etc, don't let it go!
I'd send a certified letter to them again. I'd call as well. Did they offer any explanation of how they came to their conclusion? Assuming it's a genuine ID theft and there's no reason to not believe it is, their response would anger me to put it mildly.