If I'm reading this right, it seems OP's daughter doesn't have anyone to consistently watch her during those recess periods no matter what the option is. Depending on how many recess periods there are a day, that's a lot of balancing and juggling of where to send Kid each day, and OP mentioned that recess is the teachers' duty-free breaktime, so it would be unfair and likely illegal to request a teacher on break just handle her.
I don't know if I'm crazy about using it as extra tutor time, either, unless the student is really fighting for it herself. It sounds like she needs a consistent place and person to take a quiet break with.
A few ideas that may or may not be feasible:
1. Teacher's helper or office assistant. This will take a lot of pre-planning to be consistent and not just make extra work for all the teachers, but if organized well could be useful and fun.
2. Arrange the staff so there is someone to be with her during recess breaks (without interrupting the duty-free break times) so she can have a consistent supervisor. They could sit in the library with her, plan quiet activities, etc.
3. A modification to what recess looks like at the school, which might help many other students. Perhaps another recess aide could be hired to supervise kids in an official recess-alternate spot (library, quiet area, etc). Yes, might require staff rearrangement and extra hiring, but could be well worth it for a school-wide, long-term change.