It seems like this quest might lead to a guaranteed Symbol of Avarice. I beat the game and the DLC, then looked up how to finally find the Purging Stone. Told Lapp about it and then went to the Shared Grave. In one of the rooms, there was a Symbol of Avarice just sitting there (it might have been where the Mimic used to be?) The rest of the questline played out as expected: he was in a room squatting, kicked me over a ledge, and then I found Lapp's set when I got back up.
Haven't seen anyone else mention this happening, but it really made sense to me that after Lapp realized who he really was, I found a Symbol of Avarice. It could mean that he went scrounging for treasure, or it could mean that he left his Avarice behind, which is in line with him saying he was "devoid of all worldly wants".