Q: Is it possible to update an iPod 4th gen using the beta program?

Your device cannot be updated any further. It may still have some resale value, or you could give it away.

tt2

Maybe a compatible version of the apps yo want are available:

Starting when iOS 7 was released, Apple now allows downloading the last compatible version of some apps (iOS 4.2.1 and later only)

Install the latest compatible version of an app on an earlier version of iOS or OS X

https://support.apple.com/en-us/HT201377

App Store: Downloading Older Versions of Apps

https://sites.google.com/site/appleclubfhs/support/advice-and-articles/app-store -downloading-older-versions-of-apps-ios

You first have to download the non-compatible version on your computer. Then when you try to purchase the version on your iPod you will be offered a compatible version if one exists.