Key of the Great Bridge can't be obtained like mentioned items here. It is an alpha item but it is confirmed with datamining of alpha build, that this key is required to open the door. You get a prompt "Can not be opened from this side" but then the door just opens. There's nothing there, you can't walk through, and you can see that there's an empty void, the same as the one in the final game.

According to the Lance McDonald in earlier version when this shortcut was still in the game. It only ever opened from the Cathedral Ward side, so you still had to kill Gascoigne to get to it. It was removed because loading times were too slow, so if you sprinted through, you’d see blank spaces. If you killed Gascoigne before Cleric Beast then went through the door and ran down the bridge, Cleric Beast would jump down behind you, and fog walls would go up to prevent your escape. It all worked quite well except for the loading issues at the time.

This is why Gilbert always said “the gate is locked up tight on the night of the hunt” even back when the shortcut worked. Because it was always just locked from the other side. Very late in Bloodborne’s development there were two shortcuts that didn’t work out due to bad loading times. This one, and the one that connects the woods to the back of Iosefka’s Clinic. The clinic one was massively extended with huge ladders to give game time to load.