I do think there’s a possibility that Bury somehow obtained Masonic cuff links, and Berry saw them and wrongly concluded that Bury was a Mason.
It seems to me that one of William H. Bury's characteristics was that he liked to be thought of as more or better than he was -- better educated, richer etc.
Therefore, wearing jewelry of a powerful organization would be a way of creating an image. He would not have to lie or say a word. People would simply see him wearing a Masonic emblem and interpret it to mean he was a member.
William H. Bury is one of my favorite Ripper suspects, but I keep wondering if his need to be seen as more than he was led to him trying to link the Ripper to himself . . .