CIP rules require you to collect the legal name, and use either documents or non-documentary methods to verify what you have collected. The customer should tell you what their legal name is. Then, based on your senario, review and record a description of both of the documents the customer has presented. Since they have provided documents with conflicting information, you have a discrepancy that should be resolved to the extent you can form a reasonable belief you know this persons true identity.
This was an issue I had difficulty getting across to our employees in the beginning. Do not obtain the legal name from documents presented, obtain it from the customer and then use the documents to verify the legal name obtained from the customer. In your case, simply ask the customer which name is their legal name and go from there.
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An error is not a mistake until you refuse to correct it