True minimal shoes tend to have very thin soles and no heel drop. I have Merrell Trail Gloves. New Balance Minimus series are quite popular, as are Vivobarefoot. You might benefit from a hybrid type shoe like the Nike Free, Merrell Bare Access, or Brooks Pure series. These are all pretty squashable shoes as they have very lightweight almost sock like uppers. My shoes take up less space than a single traditional running shoe and weigh in at less than 7 oz per shoe.

If you go full minimal, please read up on the proper way to transition into running in those shoes to prevent injury.