Oh MAN.....
This is a terrible thing that happens to goldfish- ive seen it personally twice, and it still makes me shudder.
the only way i have ever pulled this off is by literally poking a hole in the white membrane. the first time it was with a small kitchen knife, the second time, it was a dry fountain pen nib (it was a strange situation). its very hard to do. you are in better shape if your fish still opens his mouth normally, as they naturally do even without the membrane. i moved the problem fish into a sterile bread baking dish (a deepish, short vessel) and gently pinched his sides with my fingers (to keep him from moving) when he opened his mouth, i had to make a quick, accurate stab into the white layer of icky, and repeat it a few times to get a decently sized hole. after, i changed the water in the sick fish's tank (the people i was hired by had it in a separate tank thankfully) and dosed the water with about three antibacterial and anti fungal agents. i returned the fish to his hospital tank and watched him for a while. he kinda tried nibbling at the gravel at the bottom, which was encouraging. when i returned the next day to dose the tank, some of the white stuff had gone away, but the hole had resealed. i had to do the whole fish-in the-bread-dish thing again, and this continued for about a week until the white slime got thinner and thinner before falling out of the goldfish. i told the owners to keep the fish separate for at least three weeks and to dose it with anti bacterial and anti fungal remedies. in four weeks, they contacted me again, telling me the fish was healed.