It's a small private school and I'm the person who volunteers to keep the ukes in order (I actually started the first ukulele club at the school 6 years ago), and we've been doing this as cheaply as possible...As somebody who works at a school managing student-use equipment, consider just telling the school. They likely have a replacement on hand to give your child right away, and an arrangement with their vendor to handle repairs.
IIRC then the above is the normal solution to this problem with this brand of Uke. I don’t know how common the problem is but I feel sure that it’s been raised here in the past. A bead will doubtless be fine and is a popular way to do things, instead I use a very very small washer and put a big knot on the string’s end.Drill straight down through the top and make that one string a "string through". Use a small drill bit, not much larger in diameter than the string. Push the string through until you can fish the end out through the sound hole. Back the knot up with a bead that is large enough to spread out the load a bit. Unless there are a couple of extra inches of string length at the tuner, you'll probably need a new string.