Technically it is definetely possible to recover your LTC.
But without owning the private key (which you obviously don't because the address does belong to GDAX) you can't do this yourself.
The support of GDAX are the only ones who can help you in this case.
As ranochigo said, i also think they missunderstood your support ticket.
Its probably not the first time they recovered wrongly send coins.
Just ask them again and explain that you have sent LTC to
their BTC address.
Any thoughts what I can do if they just say they wont/cant do anything? Im just torn, killing myself over this.
Unfortunately, if they don't want to help you.. you are out of luck.
It is all up to them whether they are going to help you or not.