More or less at this point your only hope is Coinbase pulling the private key from the BTC address and importing it into your LTC wallet.
I would put the odds of that between slim and none but not at total zero. They have done it before, but it's rare.
EsPargay
I would just add on this that it would be a good idea not to say to Coinbase support that Stake sent them those coins, even if they probably know that with tracking analytics.
Better tell them that you made a mistake with sending, because I doubt they want to have anything to do with third party especially when it's gambling related.
You don't have to lie, so if they ask you if you used Stake you can say that you did.
PS
I hope you learned your lesson with this, better do double or triple check addresses confirmation before you send or withdraw any coins.