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How will the pictures be used/viewed? In an album? Displayed in a frame? Added to a smart phone?
"Portrait" suggests a formal, studio-shot photo.
"School Portrait" suggests a photo of a child in a school setting.
Preschoolers (NB-K) move around a lot, and attention spans are short.
Years ago I shot a lot of kids in activity-type settings and got the best results with a long, fast lens (135mm, f/2.5) with Tri-X film and with available light. My major objective was to get photos of kids doing something and being absorbed in what they were doing. Those photos were printed in the local newspaper. When I needed flash I used it, usually bouncing it off a wall or the ceiling. I began shooting after the kids got over their curiosity of my being there. I also got down on their level, usually sitting on the floor, and watched for opportunities to shoot.
That does not seem to fit what you're considering, but I think the basic approach would work. If I were to shoot the type photos you're looking at, I would add one step -- To cut down on background clutter place/tape some large foamcor or art boards on a couple of walls wall behind the learning centers and sit near the middle of the room.
Teachers might be willing to encourage a child to come to a designated spot, say, a puzzle, that has a white board behind it, and keep the child engaged long enough to take a few shots. This would be more of an 'environmental portrait' than a studio portrait.
This approach will take some time.
Or, you could set up a "portrait area" and set up lights focused there. Teachers could bring the children to that spot.