WTP: Negel vasser cup covers are about 30 cents a piece in the US. They work great for standard size “teplech” and have saved us some trouble many times.
But do train your kid to take the neggel-vasser out right after washing as there’s no cover I know of for the bowl. This should solve your main practical concerns.
IMHO it’s definitely preferable to follow the school’s guidelines in this case, especially if the kid is apt to tell his classmates that he didn’t have to, since his mother said so – and possibly cast both you and the school in a bad light (you for not following the letter to a T, and the school for piling on Chumras some don’t hold by).
Good luck!