The BTC deposit address I got from GDAX started with a 3. I plugged it into withdraw LTC on Binance. The same exact address exists both as a BTC and LTC address. Also, given that the BTC address is owned by GDAX, can we assume that the LTC address is as well?
Yes, they can use the private key of the BTC address to access your LTC.
This is big money. Be insistent, yet polite. It's the best way to get things done.
Admit it was your mistake and that you are willing to pay a small fee for their effort. You probably are, right?