Thank you so much.
Would such information (missing gun and badge) be released to the public? If not, can the information be discovered by a persistent reporter on his own or only through a public or press liaison person?
It depends. If the agency thinks there is a benefit to them to release the info that the murdered officer's gun and badge were stolen, they would release the info.
If you want them to hold back the info you could justify it as one of those details that are not released so that if someone contacts them about the crime, or confesses, and they DON'T know about the stolen gun and badge, then the tip is likely no good/they aren't the real killer, etc.
A badge and gun that are stolen when an officer is murdered isn't going to embarrass the department like an officer just plain losing his gun and badge. (Like leaving the gun in a bathroom, or leaving them behind at a strip club, both of which have happened). Those are the types of things the departments try to downplay or cover up entirely.
As to "what can the reporter find out." How good is the reporter? Do they have sources or friends in the police department? A good reporter should have sources that would know this kind of thing and who might leak it to the reporter. These could include officers involved in the investigation, other cops or civilian employees who heard it in the rumor mill and passed the gossip along, or even someone who was at the scene, like a paramedic or ME assistant, or funeral home employee their to pick up the body, who heard the cops talking about it.
If the reporter in your story only knows what the police agency's PR person tells them, they aren't a very good reporter. There's more to reporting than rewriting press releases and just finding out stuff from official spokespeople.