Noise reduction hood @ exhaust vent
Add a range hood damper if you dont have one already, it should close off the tube when there is no airflow.

I would still build that enclosure though. The enclosure should be made of non porous, heat/flame resistant, and corrosion resistant materials since your exhaust would likely contain lots of water and oil vapors. Some suitable materials being galvanized steel, aluminum, stainless steel, and copper.