Did the author of the Gospel of John invent the Doubting Thomas story to rebuke the followers of Thomas?

No. Had the doubting Thomas incident been falsified for that purpose, the writer would not have included the information that Thomas, following his initial doubt, knew it was Jesus with absolute belief.

John 20:28 And Thomas ed and said unto him, My LORD and my God.

Thus - no, there is no reason to accept that the author of John meant anything other than simply retelling an event to which he was a witness.