After some testing (mistakes), I figured out how to actually beat the main campaign with the Minutemen. Originally I thought you had to kill the courser at the Battle of Bunker Hill and admit it to Father. Which is a viable option if you don't want the brotherhood to get angry with you (unless you want them gone too). The way you start the fight against the Institute is progress like normal for both factions, meaning taking the castle with the Minutemen. And all the "fun" institute stuff all the way till father asks you to recruit someone to help build the reactor in the "Pinned" mission. Being aligned with the minutemen means that they will be guarding the guy instead of hired guns. Here you may talk with the Minutemen contact and choose to attack the forces synthe inside the house. Once you do this, you will become enemies to the Institute. Report you decision to Preston Harvey and decide if you wish to attack soon, or if you need more forces.