Is it possible to get this card working
I'm sure that anything is possible, but since you already talk about kernel modules, I assume you have some Linux experience. So isn't the first solution to try and compile your own LibreELEC build with any driver that is available for your 10GB card? And see what happens?
I have a feeling that the current LibreELEC developers do not have yet such a fast ethernet card in their home or development environment..