Hmm. I can't answer your question, but I don't think I would have done a chimney hood with that high of a ceiling. The very long chimney looks awkward. Is it too late to change your layout and put the cooktop/range on the window wall? Then you could get a wall chimney hood or even do an insert hood inside a hood cabinet.

Awkward placement, I would install a down draft vent if you need the cook top in that location.

Cook top should have been moved off the island. There is also to much going on between the island and the fridge wall. I can't even imagine trying to cook at the island and have people coming in opening the fridge, using the microwave or opening the ovens. Maybe your client needs to visit houzz forums.

Downdrafts are useless.

How far along is this build?

Went searching for photos. Sorry, it just looks ugly ... a stainless steel shaft hanging from the ceiling isn't a great design element.

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Sounds like you need to give the client what he/she wants, but I agree with Sabrina and LA that it is not a great layout. I'd place the cooktop on the back wall with the ovens, move the microwave (drawer) to the island, and move the fridge at the end run of cabinets on the sink wall.

Since you are building this for a client, think about how they would feel knowing you are crowd sourcing their window package and kitchen functions?

First, if the island can't be moved, then the trusses have to be moved -- either phased differently across the roof or quasi sistered outside the duct passage space. The nearby trusses will also provide a strong point for fastening the weight of this assembly hanging via the duct cover.

I wouldn't want to be the person who has to hang this. I think the soffit box will hide the inevitable "growth" of the hole in the ceiling.

I would use a down roof blower, such as Wolf, Abbaka, and Broan make, unless there is some reason for an up-blast commercial unit. With a roof blower, it will be possible to oversize the unit so that it can be effective at lower than full speeds. This reduces noise, given that there is no place for a silencer in the duct path.

Be sure that the hood itself overlaps the cooktop, and that it is high enough for good sight lines across the space. Those two requirements, combined with a 10-degree from vertical nominal plume expansion angle from the largest pans to be used, establishes the minimum hood entry area size.

A canopy hood as in your drawings, rather than the skinny poser hood styles shown in the comments by others above, should be used to provide room for baffle tilt and some 'reservoir' volume under the baffles. Example Pro Island Wolf hood below illustrates interior guts (not typically in Wolf publications) and short duct cover. This particular hood example is long enough for a 36-inch cooktop and induction wok, but shorter versions are available to fit a cooktop alone.

If a gas cooktop is to be used, some consideration for gas line installation and repair is needed. Otherwise, induction will be much safer in various ways, particularly if children will be present.

Last, I can't resist noting that the client(s) should recognize that architecture and embedded kitchen design is a systems engineering like recursive process, and nothing should be blocked from change until all aspects are defined and designed in. (See layout comments by others above.)

P.S. The Wolf above is attached by bolting the duct cover to attic structure, then bolting the hood to the duct cover.

To pile on... This is just really bad design. Move the range to a side wall where a proper exhuast hood can be installed.