inspectorD

As long as you like takin a shower with the ductwork...that would be fine.
From actual experience, I can tell you, those walls are both the back walls of the showers in the bathroom. Look at the plaster pressed through the wire mesh...and the bottom of the tub. Then the bottom of the other has the ol lead flashing at the base..on the other side of that...I'm guessin tile.

__________________
Just My
Made in the

Wuzzat?

Good eye!

I guess it's against Code to use a plumbing vent pipe as a range hood vent pipe.

Why not run a vertical 3-1/4" duct between the studs, up or down, until you can go horizontally, with maybe a 24vac booster fan at the distant end?
If Broan won't help with this, ask them for the Fan Curve
http://www.greenheck.com/media/articles/Product_guide/perf_basics.pdf
for this hood blower and use the Engineering Toolbox to calc. the duct airflow resistance.