I understand that you are receiving invites for games through GamePigeon, but when you respond to it it asks to install the app, even though the App Store shows it's already installed. I know it's nice to have your iMessage apps working well, so I'm happy to help.
Check out the information in this article to be sure that you haven't turned this app off:
Use iMessage apps on your iPhone, iPad, and iPod touch - Apple Support
Manually add or remove apps that you download
If you turn on Automatically Add Apps in the Manage tab, your iMessage apps automatically go to your app drawer when you download them. If you turn off Automatically Add Apps, you can manually add apps to your app drawer. You can also manually remove them. Here’s how:
- Open the app drawer and tap .
- Tap the Manage tab.
- Turn on the app to add it to your app drawer. Turn it off to remove it.
Thanks for using the Apple Support Communities. Cheers!