Hi there Jtolson84,
Thank you for using the Apple Support Communities! We understand you are unable to access your Game Center data via your new iPad, for a particular game. You can see it on your Apple TV's Game Center, but you also see a network error there, too. Let's see how we can help.
Let's start by troubleshooting the connection error in Game Center on your Apple TV. These steps will help: If you can't connect to the App Store, iTunes Store, or other Apple services - Apple Support. Also, try connecting your Apple TV to a different network, if available, to see if the error remains: Get help with Wi-Fi networks on your Apple TV - Apple Support. Restarting your Apple TV is also another good step to try: Restart Apple TV - Apple Support. Let us know if these steps help clear up that alert.
In regards to Game Center on your iPhone. We would suggest confirming you are signed in to the same Apple ID that you are using on your Apple TV's Game Center. Then, confirm Game Center is turned on under Settings > [your name] > iCloud > Game Center. If it's already turned on there, you should try turning it off, then restart your iPhone Restart your iPhone - Apple Support, and turn Game Center back on under iCloud.
Steps under, 'Create a Game Center Profile' and 'Use Game Center with iCloud', in this article, relate to this issue: Use Game Center to play with friends - Apple Support.
You'll also find more guidance under the section titled, 'If you don't see your gameplay data on a device', in this support article: Access your Apple Arcade gameplay data on all of your devices - Apple Support.
If you need us further, reply back with any other details and results to the above steps.