Yes, I think that is one of the options... another one has to do with bringing the fresh air into the cold air exchange of the furnace, and still another is basically what you are saying but having some sort of sensor wired to the vent like you suggested, but having a pressure sensor of sorts so that it will pull fresh air only when there is a need versus making an egg for three minutes on the range and having the kitchen flooded with cold air!
Does the hood have a variable speed fan ? That way you can run its speed according to the need
In most commercial kitchens I’ve worked in they leave the back o door open with a screen door to keep bugs out.