Salutations HnyBear69,
Let's see if you're able to manage the notifications for Game Center directly from the Apple Watch using these instructions:
"How to mute or turn off an app's notifications on your Apple Watch
- Touch and hold the top of the watch face.
- Wait for Notification Center to show, then swipe down.
- Swipe left on the notification that you want to mute or turn off, then tap the More button .
- Choose from the options that are available for that app. Depending on the app, you can mute notifications for the rest of the day, mute them for an hour, or turn them off completely."
Notifications on your Apple Watch - Apple Support
If that doesn't work, try disabling notifications for Game Center by going to Settings > Notification on your iPhone, then find Game Center in the list and turn off "Allow Notifications". If your notifications are set to mirror your iPhone, this should stop them from coming through.
Thanks.