Apple has made a mess of the new Background apps notification feature according to a well respected forum contributor who also happens to be a third part app developer. He has mentioned that developers have been not provided any information by Apple with how to handle this new Ventura feature when developing apps. Some apps seem to work Ok, while other apps keep alerting. According to this developer don't expect a quick fix...he provided information in one of his posts describing the problem of getting Apple to fix the problem...first Apple has to be able to reproduce the issue. At best, maybe Apple already knows this feature is broken or incomplete and is still developing it. Be very careful attempting any "fixes" you may find on the Internet since according to this developer most of the suggestions he has seen are just going to cause users more issues.
The only things you can do for now are:
- Uninstall the app and find another app to replace it
- Contact the app developer for assistance
- Live with it until Apple fixes the problem or the app developer release an updated app which resolves the issue
- Provide Apple with product feedback (Apple will not respond to you)
- Provide Apple with a bug report, but this requires lots of good details including how to reproduce the issue. Most people are not equipped to provide a bug report with sufficient & appropriate information for Apple to fix the issue.