This is a really bad analogy, and it's not true anyway. There are inscriptions from his period, and the insinuation that the bishop was lying reflects a colossal will to believe in something with no history prior to that bishop's lifetime.
Oh, it IS a good analogy. Tutankhamen had been almost totally erased from Egyptian history because of his family's radical religious views. Tut's original name was Tutankhenaten showing his parent's and his fidelity to the Sun God Aten. While there were hints about his reign, that's all there were. The internal turmoil from three generations of Pharaohs deemphasizing the pantheon of gods in favor of a unitary god resulted in a rabid backlash against anyone associated with it including poor Tut... who tried to put things back the way they had been. Until Carter opened his tomb and found "wonderful things" not much was known about Tut.
There were many hints about the existence of a cloth with the image of Christ on it in the First millennium. They may have been called by different names... Image of Eddessa, The Mandylion, The Veronica, they all seem to have been referring to the cloth that we now know as the Shroud. While some cloths still survive today that go by the name the Mandylion and the Veronica, they are obvious paintings. Why shouldn't it be postulated that the Bishop might be lying? He certainly had a strong enough motive. Business. During this period high offices in the Roman church were often sold to the highest bidder... and a See was a very lucrative office. Sales of Indulgences and donations enriched the coffers of the Cathedral and allowed the Bishops to live like wealthy noblemen. Perhaps Henri was not one who bribed his way to his See, but it was common. This was a time in which there were two popes because of such shenanigans.
You have no trouble ascribing venal motives and unscrupulous actions to a French knight whose well documented history is impeccable (Standard bearer for the King and author of the Code of Chivalry) and who almost bankrupted his family supporting a Church to house his most prized possession, the Shroud.