Help with floor joist cutting

Hi, new to forum and need some help with installing kitchen vent ducting. I need to install a kitchen hood for an island cooktop. The bracket to hold the hood will straddle a floor joist. The bracket needs to be recessed up in between the joists and I was thinking about doing a plumbers box but the vent needs to go through the box...so I'm kind of stuck there. I'm wondering if the header that has to have the vent going through it if I could sister it with either one or two additional pieces or use a flitch type construction for it, if that would give sufficient strength? Even putting additional braces between the cut joist and the adjoining trimmer for the vent run. The CFM of the vent requires an 8" round duct. The joists are 2X10 and one of the planned trimmers is already sistered. So the header on the joist closest to the outside wall would have to a have the hole cut into one half of it.

Thanks for any help or ideas.
Yank