I've worked for a very similar program for the last few summers. Mine is a new play festival, a professional theatre, run by a major university with ties to the local LORT theatre. We do 3 plays in six weeks and its pretty much non-stop action. The first play finishes its second week of rehearsal in tech, and then the schedule goes: Play 1 runs for a week, Play 2 runs for a week, Play 3 runs for a week, All 3 plays run in repertory for a week, culminating with all three shows back to back to back on the final day.
The first year I did it, I was one of two stage managers for the 3 shows. I stage managed the first and third show, and it was insane. The second year they realized that they absolutely needed to have 3 stage managers. I was the PSM, coordinating both of the other 2 shows, and stage managing the 3rd. It worked a lot better. Each year we'd get 2-4 college students as assistants. They were part of an apprentice program that the university also ran, but their focus was more acting, so whether they were any good at ASMing was hit or miss. I wish that we'd had interns specifically for stage management.