As the site had been hacked and they have no backup of that database, I do not think it is possible for them to refund base on the information of deposit from a wallet. Think it yourself what will happen if same claim occurred by multiple person? You may think it is not possible as you are a legit users but there are a lot of people who will try to take a chance for the exploit of the site. I think for now they have no alternative to verify except the email which was on the receipt on the email.
Think it yourself. They have the wallets, all transactions are onchain and no multiple person deposited from the same address. Anyone trying to exploit repayments would need access to your address to take your funds if deposits was sent back where they came from.