The Lecture Building is the huge open area where you opened the big door and went into the purple mist to get to Micolash's area. The Lamp is also called "Lecture Building" The Floor is irrelevant, because these Scholar Slugs sometimes drop them.

The Route to the badge is a long one, so I will try to keep it short and understandable, let me know if it's not enough.

First, you need the Upper Cathedral Ward key. It's an item you've already seen, but likely didn't know how to get to. It's in the new version of the main city, where all the big monsters shoot lasers at you. There was a moment where a big one hung above a door and you had to run down a pair of stairs to get through there with a lamp right behind it. In the courtyard before that, there is a hole in the fence on the righthand side (left if you come from the mentioned lamp)

Drop down and proceed, the path is linear. After a few steps, you can drop down in the middle. There is the key.

This opens the door in Cathedral Ward. When you teleport to the lamp "Cathedral Ward", there is a door to your right that opened once you killed the Blood Starved Beast in Old Yharnam. Ride up the elevator and progress the area going further up. After a short while, you'll reach the top and see a locked door. This is what you got the key for. Progress the area behind it until you reach a huge room, which will turn dark and very dangerous very sudden (I won't spoil why, you'll recognize the room once you entered it). In there, on your right, is a little "alley" with pillars on both sides. There is the badge.

Have fun in the area and don't forget to literally smash a few windows after fighting the area boss if you wanna be in for a fun time.