iMessage notifications won’t go away

Hello cg21734,
Thank you for using Apple Support Communities!
I understand from your post that the badge app icon on the Message app will not go away even though you have no messages. You have done some great troubleshooting so far. The next steps I would recommend would be to backup and reinstall the software.

The following article has the steps on how to backup your iPad:

How to back up your iPhone, iPad, and iPod touch - Apple Support

By using the steps as listed from the following article you can reinstall the software on your iPad:
If you can't update or restore your iPhone, iPad, or iPod touch - Apple Support

Put your device in recovery mode and set it up again

1. Make sure that you're using the latest version of iTunes.
2. If iTunes is already open, close it. Connect your device to your computer and open iTunes. If you don't have a computer, borrow one from a friend or go to an Apple Retail Store or Apple Authorized Service Provider for help.
3. While your device is connected, force restart it with these steps:
   • On an iPhone 6s and earlier, iPad, or iPod touch: Press and hold the Sleep/Wake and Home buttons at the same time. Don't release the buttons when you see the Apple logo. Keep holding both buttons until you see the recovery mode screen.
   • On an iPhone 7 or iPhone 7 Plus: Press and hold the Sleep/Wake and Volume Down buttons at the same time. Don't release the buttons when you see the Apple logo. Keep holding both buttons until you see the recovery mode screen.
4. When you see the option to Restore or Update, choose Update. iTunes will try to reinstall iOS without erasing your data. Wait while iTunes downloads the software for your device. If the download takes more than 15 minutes and your device exits recovery mode, let the download finish, then repeat these steps.
5. After the Update or Restore completes, set up your device.

Best Regards.