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Check the spending power tool on the website to see if they will allow the charge
Has he tried checking the "Check Spending Power" feature on the Amex website?
I believe it's based on a rolling 6 months of use.
Then I would suggest pushing a payment to the Green card from his bank to have a negative balance so the charge goes through. If this is not viable then try customer service to see if they can work with him on getting it approved
Call the number on the card and speak with a Customer Care Professional. They can authorize a charge in excess of the current limit.
I thought the internal limit for new accounts was $5300.