NO, you don't have to.
In US, most tournaments are open tournaments. They normally have ceiling rating, such as U1300. If you are above 1300, you can't play in this section. But most don't have floor rating, so you can play up in any higher section you want to. Only a few tournament sections have floor rating to guarantee the strength of the sections so they can attract more strong players. In this case, they have to specify the floor rating in TLA announcement, otherwise the tournament director can't enforce the floor when players arrive.
For a scholastic player, most time s/he wants to play up to have more difficult challenges, as I suggested in another blog article. You, as a parent, have the ultimate decision right. No tournament director can force you into a lower section unless the floor is pre-announced. They can suggest, but they can't force. Please remember that tournament directors most times are not strong players, not coaches, and not knowing your child well. S/he makes suggestion only based on the standard view of rating, i.e. if you are not above 1300, you should play in U1300 section. Or s/he may try to balance the participant numbers among sections. You, yourself with your child, have to decide what's best for your child.
In addition, picking a lower section and trying to win it is not good decision in psychological perspective. Being the top seed, your child will have tremendous pressure, and could easily blunder under such high pressure.