[The Lucas]

You don't need a high school's support to have a team, we don't (as gnitcixe said). We used to be an Explorer Post (Boy Scout related) but now we use "MOE Robotics Group" instead of a school. You don't need to go that far. FIRST only requires that the students on the team attend a high school. Our high schools provide nothing for the team and are completly unassociated with the team. The only thing the schools do for us is let us out of school for competition, not like they could stop us if our parents just wrote a note instead.
However, what you need to do is change your team's info when you re-register to remove the school and put your sponsor in its place. You may or may not be able to do this, so if all esle fails create a new team with the sponsor (it will be four digits) and the students from St. Clemments or any other school or home school can join. It is an out-of-school activity so the school does control its students or teachers out-of-school. Unless, of course this is a boarding school, where the students would need permission to leave for meetings.