As a lifelong vegetarian, I'm concerned. Would Sai Baba have made me eat Prasad containing meat?

You need not be concerned. Sai Baba was very mindful of his devotees' practices. Chapter 38 explicitly states that for those who "never had such association in their entire life, He would not let them even touch the meat." Baba never committed the rashness of creating a new craving for it through the Prasad. He only served meat-mixed food to those who were already accustomed to it, showing his respect for individual dietary habits and conventions.