This looks like a very early release and may not work/may not work well. That being said, articles from redmondpie.com, and idownloadblog.com outline a method for sideloading Kodi onto an AppleTV. With Kodi installed you can use samba or similar to share content from your Pi to Kodi on the AppleTV. You are, apparently, going to need a Mac running XCode 7 and a USB C to USB A cable.
As posted on idownloadblog.com:
Step 1: Download the latest nightly Kodi.deb file, the iOS App Signer,
(save both to desktop and unzip the iOS App Signer) and make sure you
have Xcode 7 installed.
Step 2: Connect your Apple TV to your Mac using a USB Type-C cable.
Step 3: Launch Xcode → click Create a new Xcode Project → select
Application under tvOS → Single View Application, and click Next.
Step 4: In the Product name field, insert a unique name like
KodiAppleTVforYourNameHere and click Next → Create
Step 5: Under the Team drop down box, select your personal team, and
click the Fix Issue button to resolve provisioning issues. You will
need to ensure that you are signed into Xcode with a free Apple
developer account. You can learn how to do that using this tutorial.
Step 6: Ensure that your Apple TV is selected in the destination menu
at the top of the screen. You can also click Product → Destination,
and select Apple TV.
Step 7: Minimize Xcode and open the iOS App Signer.
Step 8: Click Browse on the iOS App Signer and select the Kodi.deb
nightly file from the desktop.
Step 9: Select your Apple ID in the iOS App Signer’s Signing
Step 10: Select the Project name/bundle identifier you just created in
Xcode in iOS App Signer’s Provisioning Profile box.
Step 11: Click Start in the bottom right-hand corner of the iOS App
Signer, and the .IPA file for Kodi will be built and saved to the
Step 12: Open Xcode → Window → Devices and select the Apple TV
Step 13: Click the ‘+’ sign and select the .IPA file from the desktop
and click Open. Kodi will now be deployed to your Apple TV
Step 14: Once finished, you should see the Kodi app on your Apple TV’s