We are currently getting ready to deploy 8.1 tablets (Surface 3 Pro). Our deployment includes Office 2010. We expect users to use OneNote 2010 and save all notebooks to local/network storage, they should not be using non-company storage for any data. This means we have to disable use of OneNote Modern. Additionally the button on the Pen automatically launches OneNote Modern so we need that to either be disabled or preferably launch the correct version of Outlook.

You could disable it using AppLocker, so even if user install it or if it is already installed, they are unable to run or use it.

Some suggestions:

Block the use of Windows Store entirely. uninstall (unprovision) the OneNote modern app.

Use AppLocker through domain GP; (this needs Windows Enterprise edition to have AppLocker)
https://social.technet.microsoft.com/Forums/windowsserver/en-US/51ea5d09-3b78-4830-a67f-a153c5c2e559/how-to-manage-windows-8-store-apps-using-app-locker?forum=winserverGP

https://4sysops.com/archives/controlling-windows-store-apps-in-windows-88-1-with-applocker/

http://blogs.technet.com/b/canitpro/archive/2013/06/10/step-by-step-controlling-access-in-windows-8-with-applocker.aspx

You can configure the pen so that top-button will open OneNote2013 instead of OneNote modern app. (there is a setup wizard, but I can't remember how you get to the wizard. Maybe you just search for "pen" in Windows)

EDIT: https://www.microsoft.com/surface/en-au/support/touch-mouse-and-search/surface-pen-default-onenote?os=windows-

Thanks for that. I'll think about the AppLocker method, or possibly unprovsion it. Unfortunately we are currently stuck with OneNote 2010, I saw the 2013 fix, but couldn't see a 2010 one.