How to install sprung-loaded backdraft damper into existing wall vent.

Would like to install a Neverest sprung-loaded backdraft damper (photo 1) into exterior wall vent of downdraft KitchenAid gas stove Model KGCD807xss01. The Neverest assembly is designed to slide-fit into existing ductwork.

How can we achieve this given the following information?

Thank you in advance for your guidance.

:::

Background Information

The hooded vent cover (photo 2)

is fitted with a gravity-operated flap (photo 3)

that vibrates very loudly with the wind. We have resolved the noise by putting some foam on the lower edge of the flap (photo 4).

However, perhaps as a consequence, the cold return air flow through the downdraft is significant. In addition, the cupboard below the stove has become a mini-fridge, there being so much cold down-draft metalwork under there.

On removing the 4 wall vent screws, we cannot extract the wall vent hood. Reaching into the ductwork, we can feel a transverse seam about one foot in. At the hood itself, the ductwork is bent over sealing in the hood (photo 4).

The new Neverest damper unit (photo 1) will not fit up and under the hood to be slid into the ductwork even if we were to snip off the current gravity flap.