I can see your conundrum . . . the hood is just a bit on the shallow side and the narrowing of the chase as it goes to the ceiling means the body of the unit comes into play as well. It's possible that the Best by Broan, model PK22 would fit. It might be best to consult an appliance store where they can handle the installation and check for fit before you purchase. Otherwise, I'd check to see if the cabinet manufacturer has a suggested unit that would fit. Good luck!

i'm not sure of all your measurements, but if you look at some of the insert hoods, they might work. most are 18" deep.

however, this one is 34x15...if you can find it in stock (saw it on HD site)

Ancona Chef insert

Internet #303780984

Model # AN-1313

Was the hood custom made or a factory cabinet? Usually a hood is custom built according to the specs provided with the hood insert. Of course it's spilled milk now but ideally, that should've been bought/decided on before the cabinets. If it's a factory cab, then find out from the company whose specs they used to design the hood.

Reverse order of operations.

And that hood is mounted too low. It’s a fire hazard. So it has to be redone anyway with something new that actually has some capture area.

Start with the insert. Bigger than anything you have. The hood gets custom built around it.

Love the backsplash tile. Can you provide details? Also where did you end up getting a vent from? I'm having a similar issue.