Ya they gotta be the worst cops ever if they couldn't determine he didn't die a day earlier and not in a shootout. And why a shootout when he was already dead?
shareI'm thinking perhaps they wanted to protect Kate, who was traumatized enough at that time, by saying it was a shootout with the police instead of announcing she killed him.
Edited to say: In thinking about it, I'm sure the police went with that story so it wouldn't be known Kate killed him. She probably found the key to the basement in his pocket, let herself out and called the police for help.