Badami, from the Cricket Club of India, compiled small breaks of 31, 38 and 51 in the three frames to wrap up a 55-41, 53-52, 61-9 second round victory.
Amol Abdagiri of Pune led the charge of outstation cueists into the third round.
Puru Dembla from Agra and Siddhesh Mulay from Chiplun, Maharashtra, were the others from outside Mumbai who won their second round matches to progress to the next round.
Dembla sidelined local player Divesh Gangwani 3-0 (52-43, 59-46, 43-30), while Mulay fought back after dropping a frame to overcome Nikhil Ghagde of Railways 3-1 (38-56, 68-61, 74-22, 67-37).