Honestly I can believe either scenario.
The timing is also suspect in my opinion. Why wait until the known balloon is finally shot down after traversing across the entire country for days? Why isn't the knowledge brought forth immediately? Someone had to come up with the claim and perhaps even provide the cover story that makes it seem legitimate. That's how the CIA operates after all. It gives them plausible deniability. They hadn't planned for it to be spotted, so they had to create what was needed, because it didn't exist already.
Police look for people who change their story when interviewing them regarding a crime. I'll leave it up to you to determine if that method is a valid one or not, but that is what the police do actually do to determine the honesty of the person being interviewed.