If I could get a hold of the private key of the address I sent my BCH to, I could easily recover them.
Kraken does not offer a wallet service, you do not hold your own keys.
They use your deposit to pay other withdrawals, so they can't just give you your deposit address private key... That would result in you being able to both spend your balance on kraken plus being able to send an on chain transaction on the blockchain, essentially "double spending" but not in a conventional way.
There's a good chance that you won't get your money back.
however the BCH shouldn't be lost because kraken does hold the private key for the address that you accidentally sent to and if they were bothered then they can easily recover your BCH by importing the bitcoin private keys into a bitcoin cash wallet. But tbh, they're not going to go to these lengths just to restore an accidentally sent BCH or two.