As the parent of a child with tree nut/peanut allergies (who had to go to the ER two weeks ago with a reaction), I'd be interested in knowing about nut-free bakeries in general, kosher or not. I've been baking cookies at home the last two weeks, to have control over ingredients, but, as you know, schools in these complicated days won't allow homebaked items to be brought to share with the whole class.
Maybe thinking of alternatives to baked goods is the best way to go. You mentioned ice cream -- but there could well be a lactose-intolerant kid in the class.

Popsicles? Kind of messy, I guess. How about candies such as Smarties or Gummy Bears? There are those great Japanese gummy candies that would be a real treat. Or, something I've seen suggested on websites about allergy problems is to give out non-food treats instead: stickers, for example.
Good luck with this, as well as the birthday cake project!