Midland makes a .410 "folder" based on the Beretta 412 design also copied by Yildiz. The recoil of the Yildiz using the 13/16 oz. load or the 370NM might have a tad more recoil than a 7 year old would feel comfortable shooting, but you haven't indicated the load weight you are using. The Midland is moderately priced and is 4.85-.9 lbs. compared with the Yildiz 3.185 lbs. The Midland also has a styrene stock and has detachable lengths to increase or decrease LOP. It also has a small recoil pad. The Midland's weight compared to the Yildiz is the result of a non-scaled action made of steel. The same action can interchange all gauge barrels. Because of this, the 12 gauge is the lightest as more steel has to be put in the chambers in the small gauges to center the primer against the firing pin. The action of the Yildiz is anodized alloy. I prefer the lighter weight of the Yildiz, but a small child might not be comfortable with the recoil. A lighter gun has more felt recoil than a heavier gun shooting the same round. However, practicing with lighter rounds and doing the switchero with the TSS round at the moment of truth is also an option. Of course, there are the external hammer actions to consider by NEF, but in the hands of a child, the potential for an accidental discharge lowering the hammer on a loaded gun is a consideration. The Yildiz and Midland have internal hammers with a tang safety. The Yildiz is $130 at Academy (no shipping) and the Midland, slightly more.
Here's a Midland link:
https://www.shootmidland.com/collections/all?constraint=410-gaugeAcademy:
https://www.academy.com/shop/browse/outdoors/shooting/firearms/shotgunsAnd the last few pages of the below old thread detail my efforts in modifying the Yildiz .410 for turkeys:
http://oldgobbler.com/Forum/index.php/topic,26715.0.html Gil