The biggest and most important element here is how are you going to transfer the fresh air in, due to the demand from the fire you will need considerable air moving in and out to keep CO not a problem. I would absolutely have a CO detector. The other thing is the humidity, ie you will need the air circulating or you will end up with a sauna... a couple of examples, houses in AK (and in general today are built too airtight) are pressure tested to design the fresh air make up size and they include a highpoint exhaust fan the turns on based on humidity to keep surface mold from growing. Example two is in houses with large BTU devices burning inside (large BTU gas ranges or gas furnace/water heaters), have fresh air intake make up devices to keep CO down.
Many codes require an automated intake fan with a design CFS on input air vents to turn on when the exhaust fan is turned on. These systems use to be $$ but have dropped in price considerably. I would suggest a similar venting system, ie a fan powered make up air along with an exhaust fan with hood. Then when cooking you turn on the exhaust fan and the intake fan turns on to make up the air.