I would get a purpose-built hanging hood for this and here's why: Cooking smoke contains grease which can collect and catch fire. Commercial kitchen venting systems must have allowances to handle that by code. Some just use a re-usable washable filter while others also have a drip catchment system, but all will have some method of preventing a flue fire. While speaking of code, many places do not allow a barbecue grill to be placed under a roof structure that is attached to a dwelling anymore. It is now code in my county, but not an adjacent county so as always check where you live first.

If you want to fab something yourself I'd at least build it around a grease filter made for this application and stay with the CFM range that filter was designed for. I'd also maintain clearances to combustibles required for a similar stovepipe just in case something did go wrong. Better a plate of barbecued ribs without a side of fried house!