Coinbase is like a Bitcoin (Cash) "Bank". You have an account with them, and they manage your money for you. You don't manage your money yourself.
This means that they, and only they, have access to the Private Key needed to recover the Bitcoin sent to your Bitcoin Cash address. No one else can be of material help (other than offering some advise).
Technically, what they need to do is very easy. All that needs to be done, is for the Private Key to be exported from the Bitcoin Cash area, and imported into the Bitcoin area.
However, Coinbase has a very complex setup, and what is easy on a small scale can become terribly complex at the scale they are operating on. Also, I don't believe any one Coinbase employee is able to access Private Keys. They are in a securely sealed environment, if you will. This is great for reducing the risk of thefts (even inside jobs), but not great if you actually WANT to have access to it.
There isn't any reason they couldn't do it. It may just be prohibitively expensive for them to do it.
I'd recommend holding your own Bitcoin Cash in your own wallet. Ie, you control your private keys. If you have anything from $1000+ consider getting a dedicated hardware wallet such as a Trezor or Ledger.
For Bitcoin, I used to recommend the same. But with the network fees as high as they are, I'm more inclined to leave funds on an exchange if you want to be able to sell them in the semi-near future. This has lots of risks (ie, exchange getting hacked, going insolvent or committing theft themselves). But transferring also has high costs (2x high miner fees), and risk (ie, time lost when trying to send a transaction - are you even able to spend the coins in the future?).
Excited about the potential of Bitcoin Cash in the beautiful country of Belize.
Developer of the RegisterDocuments.com Document Registration Service (using the Bitcoin Cash blockchain).