A suggestion from a parent with kids in private school.
Buy professional labels, they are inexpensive, are not scratchy and look good. You can buy sew in or iron on ones.
If the kids have a unique last name, just use the last name. No point if buying labels from Johnny, Susie and Kimmy, if no one else in the school will have the same last name.
Two changes of clothes per child per day is nuts. My kids only have two uniform bottoms, no way I would pay for an extra set of clothes that will never be worn.