How can I install a hood exaust vent with a joist in the way?
I would cut your &quot;Top Support Plate&quot; (as it's called in the picture you linked in one of your comments), as you suggest in the picture of your question.
Attach a metal 6in 90 to drop down (as I suppose you were going to), and then 6in ducting down and attached to the vent unit, all inside the shrouded box.
There seems to be enough distance down to allow a minor slant to this portion of the vent, form the off centre exit point at the top plate, down to the fan unit.
And attach the plate to the joist for extra strength.