V.B.
I don't have this issue since they don't nap at my daughter's preschool, so she only has her backpack and lunchbox, but I can see your point! They do make nap mat covers that sell online that may have a handle. I would just google nap mat covers and see what comes up. I was looking into them too, but since I have to walk my son into MDO (they don't do carpool before pre-school age), I just carry it all myself and didn't want to spend the money on a fancy cover. I hope it helps and you're able to find something that works for her. It sure is a big load of stuff!
Does she really need a binder versus just a simple folder? That seems excessive for a 3 year old! They sure start early, don't they??