By analogy, when a family celebrates a child's summer birthday at home, does the child get the message that this is being done because the child couldn't handle not being celebrated in school (as may be done for those children whose birthdays fall during the school year)?
Huh? No, because a birthday would be celebrated at home anyway. If one insisted on having all one's child's classmates over just for cupcakes and the birthday song (apart from a party) and then sent them home, that would send the message that you thought he/she couldn't handle not being celebrated in school. (It would also be totally weird.)
I think ice cream for both kids sounds great, Sweetie.
