My grandchildren do not have play clothes and school clothes tho I remember having them when I was in grade school. Kids play just as hard and get just as dirty at school as they do after school. I suggest that you're creating more work for yourself than you need to have. More laundry as well as the difficulty of telling and keeping them apart.
I also suggest that if the grandparents can't tell the difference then there isn't a big enough difference for it to matter.
I suggest that we used to differentiate between play and school clothes because life was different back then. Kids had less "dirty" activity at school and washing machines were very basic and kids had fewer clothes. My mother did laundry once a week in a wringer washer. I wore 2 outfits to school for a week and one set of play clothes that I wore for a week.
My grandkids in kindergarten wore the same clothes after school as during school and then had a new set the next day. That meant only 5 outfits to wash/week. Much easier. Using your method it sounds like you'd have 10 outfits to wash and keep segregated.