This script will work with any application that has a “document one” open. I added a delay to the code which should give you enough time to bring your application with document one to the front.
You can also tweak the code to target Adobe Acrobat instead of front most application
delay 10 -- Gives You Time To Activate Whichever Application You Want This Code To Work For
tell application (path to frontmost application as text) to set the clipboard to (get path of document 1)
Before realizing that you came up with your own solution, I came up with another solution that works.
tell application "Adobe Acrobat Pro" to set the clipboard to the POSIX path of (get file alias of active doc)
This works for me using the latest version of Sierra and Acrobat Pro 10.1.1.33