Is it permissible to give charity equal to the value of a debt owed by me—whose creditor and whereabouts are forgotten—in order to pay off the debt on behalf of the deceased?

You are obligated to diligently search for the owner of the shop to deliver his right to him. If you do not find him after expending effort, then give the amount in charity on his behalf. If you find him later and he approves of the charity, your father's obligation is absolved. Otherwise, pay him his right, and you will have the reward for the charity.