The discharge status of the shooter was a “bad conduct” discharge. That covers a myriad of undesirable circumstances As far as I can tell at the moment the Federal Law says the reporting is about people with known mental illnesses. As you can see there isn’t any clear cut determination that the USAF was not in compliance or required to report anything unless there is additional circumstances involved. There may not have been anything of record that says he was mentally ill or the USAF knew he was. Conversely there may have been something known about his mental state but to jump to a conclusion that this is an existing fact that the USAF screwed up is a bit of a jump at the moment.
You are looking at this all wrong. Here's the bottom line.
While in the Air Force, he was convicted and imprisoned for committing domestic violence against his wife and child. There's this thing called the "Lautenberg Amendment" that prevents people convicted of such crimes from possessing firearms. Being in the military, once you're convicted of DV and no longer able to possess a firearm, they look to separate you in most cases. I remember 10-15 years ago this was a very big deal in Army organizations.
Once he was separated from the AF his separation paperwork should have at the very least mentioned Bad Conduct Discharge for Article 128 (Assault). It has nothing to do with him being mentally ill. He fell under the "Lautenberg Amendment" and should have been processed accordingly.
The Air Force screwed up. There's no tip toeing around that. Someone is going to have to answer for why that lunatic's information wasn't processed correctly.