They’re pretty standard parts so don’t feel like you need to go with Fantech in particular, feel a little like I accidentally shilled for them a bit, just happened to be on their website (AFAIK they do make good stuff though). Glad that it helped and best of luck :)

The switch would be something like this. I think differential pressure between the inside of the duct and your kitchen would work.

and any “normally closed” damper would work. Wired up such that when your range hood is switched on, it also switches the motor in the damper on so that it opens.

The simplest thing would be a pressure or airflow sensor hard-wired to the switch for the hood itself.

Also consider instead of a 1 way valve installing a damper that only opens when your range good is turned on.