You are both basically correct in your concerns. There is a high school rule that limits each wrestler to only one bracket. Since the Kids Division rules follow the high school rulebook except when modified, and since there aren't any Kids modifications of this rule, you cannot enter a wrestler in two weight classes, or two age divisions, at the same tournament.
The only exception comes in the case of split tournaments. Because these tournaments reach a conclusion, followed by a new start, they have been determined to be two separate tournaments for the purpose of this rule. Therefore, a wrestler could wrestle one division in the first session, followed by another division in the second session. This is not recommended by the Kids Executive Council, which has expressed concerns about wrestlers competing above their natural age division; however, it is allowable.