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I believe according to the pictures that I saw, the stiles are 3.5” wide. See the picture below (stiles have blue painter’s tape on them to protect them)
Thank you
I would simply stick with a vertical rod exit device meant for a narrow stile door.
The Corbin Russwin ED7400 as seen in the link below.
You probably need to come up with a special bracket to mount the top strike to but at least this way you'll comply with code:
Download File
If the current course of action is insisted on, you could install the Hager 2-679-0620 Power Transfer as seen here: