There is a reason they put all the warnings on deposit address screens on the exchanges... There are a number of coins that use similar address algorithms and you can end up creating a so-called "cross chain" transaction...
Which, essentially just means that you sent funds to a valid, but unintended address. Sometimes you can recover the funds, sometimes you cannot. It depends on the currencies involved, the algorithms for converting private key to public address and who "controls" the private key.
In this case, I believe the OP is out of luck as, while it was a BCH to BTC address situation... He doesn't have the private keys as it was an exchange based deposit address and they seem to have a "we don't help for incorrect deposits" policy