I think its ADPi (and not because I am biased!). Most of the clues are easy enough to interpret to ADPi:
One AI so far
Three letters
Not a "specific shape" pin
Founded in the 19th century
Already mentioned
The St. Patricks Day one is the catch. I think it works because of the McDonalds Shamrock Shakes. I remember during recruitment the PNMs would watch a movie about the Ronald McDonald House while working on the philanthropy project. The part that stands out the most was that the proceeds from Shamrock Shakes used to be (maybe still are) donated to the Ronald McDonald House. I'm not sure Lake would have seen this movie (probably not
) but she may have known the connection. Just a thought...the Claddagh hands is a good reason as well. So my guess is ADPi!!