I sent BCH to a BTC wallet on a website. Coinbase is useless and won't give me any private keys, nor do I have support from the website I sent it to. Is there a way to get a miner to help me on this or is it a lost cause?
Sorry to hear that.
Please keep in contact with support at the website you sent to. Please also keep in mind that they may have a number of very legitimate reasons (e.g., security concerns, or maybe their method of storing coins) for not attempting recovery of the coins.
Do you mind if we ask what website it was? Is this a web wallet of yours, an online account? Or were you sending BCH to pay for something on, or through, a website?
If you accidentally send BCH to a BTC address (or BTC to BCH) you may be able to recover those funds if you have the private key for the address sent to (the keys are the same on both chains).
It sounds like (based on what you've stated) you sent BCH from Coinbase. If that's accurate, Coinbase is not in a position to help you (they have no control over the private key to the address you sent to).
Whoever owns the address you sent to are the only ones who have the ability, or are in a position at all, to help you.
In the mean time you can look up, and keep an eye one, the address you sent to here (you should see your BCH):
https://explorer.bitcoin.com/bch