You do not owe the Post Office your time and effort to make up for their mistakes. Depending on the value of the item and the payout made, there's a good chance the time you would have to spend "correcting" this should be valued a lot higher that the payment. If they both delivered the item and paid out an insurance claim, they made a mistake and they need to do the work of correcting that, if they care about it.
Keep the money set aside to return if they come asking for it up to whatever the statute of limitations for a claim against you would be, and return it if they contact you with a simple procedure to do this. Otherwise, don't burden yourself with it. You have better things to do with your time. If you haven't cashed/deposited the check, you might just be able to refrain from doing so, but then it will probably end up getting handled as unclaimed property belonging to you many years down the line.