Thank you for your post.
It sounds like you are almost answering the question yourself!
In your script, you can switch to the layout that has the Google Map in the web viewer, and then follow it with a Print script step and return to the previous layout.
For example, assume your original layout is called "Workorder", and you want to print out a layout with the web viewer entitled "WebViewer". Your script may look like:
Go to Layout [ "WebViewer" ]
Print [ Restore ; No dialog ]
Go to Layout [ "Workorder" ]
To stop the user from interrupting the print step, you may want to include "Allow User Abort [Off] " before going to the WebViewer layout.
This should get you pointed in the right direction.
Let me know if you need clarification for any of the above steps.