What does a letter-of-credit issuing bank do if the buyer fails to repay after the credit period ends?

This comes under the purview of the risk assessment that was done prior to the issuance of the LC.

Most banks will prefund the LC and not work on credit (the terms credit can be misleading here).

For other clients, the LC would be subjected to the client's credit rating, and if the client defaults on the LC, whatever the bank has in collateral would then be used to off-set the default, and possible legal action or lien on wealth.