If I can only upsize the duct for half the run, does it even help out? I guess its a trade off as to whether Id rather cut a larger hole in that rim joist than I need to or not. The hood itself does only output from a 6 inch vent, I have checked.
What is the rim supporting? Usually, if there is roof or wall, it is sitting on plates (sill plates). Sill plate, single or double, can span one joist bay, about 14". That means you can remove the entire block of rim lumber from that one bay. I am guessing but I think you'd be safe to cut out a square 9x9" and use bigger duct. This is worth calling in a pro (if you're not sure) than regretting the capacity and noise.
For fyi, search wall framing. Load bearing 2nd story wall studs do not have to line up with studs underneath - for 2x4 double sill and 16" oc wall framing.