I do not unerstand why your credit card, Barclay's, should pay you and ths inherit your headache with the travel agent.Alitalia refunded the travel agent on 12/16/20. It was a full refund. The travel agent has not contacted me or given any kind of refund. They are now simply $1100+ richer. Even with Alitalia's response to the DOT showing that a full refund had been made to the travel agent, my credit card will still not refund me. This is absolute BS.
A long time ago (before 1975), a credit card company would have no responsibility in this dispute. None. (I have a long story about how I was successful in a dispute before 1975, and how I worked on a change in the law, but it is not relevant here.) After 1975, a credit card company was obligated in certain circumstances to give its customer, you, a refund, if you did not receive the goods or services you ordered. That obligation runs for 60 days after the date of the statement on which the disputed charge appears. That time has passed in OP's case. s
Sometimes, a credit card company will accept a late dispute and simply contact the seller, and get a refund. Sellers are concerned with their level of disputes as it affects their relationship (and pricing, reserves, etc.) with their bank, MasterCard and Visa. But I'm not aware of any bank or credit card company who will simply refund when not legally obligated to do so. Some won't even process a late dispute.
Thus, the angle I see OP can play is to go back to Barclay's, and report that Al Italia has paid the OTA who processed the original charge, and request Barclay's assistance in getting a refund. If the OTA values its relationship with the banks and credit card companies, it may make the refund. If it doesn't care, or spent the money on keeping the lights on, or whatever, it won't. OP will be out the scant cost of the few electrons needed for the new e-mail to Barclay's. Remember, this situation is not Barclay's fault, and you need to enlist the leverage they bring to the table.
And keep in mind that even if Al Italia refunded the money, Greek laws have changed, and the OTA may be allowed under Greek law to keep the money and issue a credit. The Greek economy was hurting before the pandemic, and the lack of tourists and the money they bring has had a devastating impact on the Greek economy.