You should be able to generate a prepaid Fedex Airbill as a PDF and email that to the TA. The TA can either stick the glasses in a Fedex box and use the label you have sent or, if it does not stock those supplies, arrangements can be made (for a fee) for Fedex to come pick up the glasses.
You will have paid on a US$ credit card, the card number will not appear on the Airbill and everything will appear on your bill. It will cost a minor fortune, so you should evaluate whether it is worth the cost, but it is doable.
The TA should include a note which you dictate in English and Arabic explaining in 2-3 short and simple sentences what the item is and how it comes to be shipped. E.g., "our customer left his glasses at the hotel and we are shipping them to him as a courtesy" (this covers both exit formalities for Egypt and CBP for the US).