Social media platforms have become pivotal for projecting national identity and soft power in an increasingly digital world. This study examines the digital manifestation of Taiwanese gastrodiplomacy by focusing on bubble tea -- a culturally iconic beverage -- leveraging a dataset comprising 107,169 posts from the popular lifestyle social media platform Instagram. Including 315,279,227 engagements, 4,756,320 comments, and 8,097,260,651 views over five full years (2020-2024), we investigate how social media facilitates discussion about Taiwanese cuisine and contributes to Taiwan's digital soft power. Our analysis reveals that bubble tea consistently emerges as the dominant representation of Taiwanese cuisine across Meta's Instagram channels. However, this dominance also indicates vulnerability in gastrodiplomatic strategy compared to other countries. Additionally, we find evidence that Instagram suppresses bubble tea posts mentioning Taiwan by 1,200% -- roughly a twelve-fold decrease in exposure -- relative to posts without such mentions. Crucially, we observe a significant drop in the number of posts, views, and engagement following Lai's inauguration in May 2024. This study ultimately contributes to understanding how digital platforms can enable or disable gastrodiplomacy, soft power, and cultural diplomacy while highlighting the need for greater algorithmic transparency. By noting Taiwan's bubble tea's digital engagement and footprint, critical insights are brought for nations seeking to leverage soft power through gastronomic means in a politicized digital era and researchers trying to better understand algorithmic suppression.