You don't have 'reasonable' doubt, Jabba. Reasonable doubt is where you can point to a clear process by which all three laboratories, including the two which blinded the samples at the final stage, could have made the same dating error. Reasonable doubt is where you can point to an actual flaw in the process of sampling, processing or reporting of results. Reasonable doubt is where the alternative theory is actually tenable without resorting to magic.
What you have is unreasonable doubt, fostered by your intense desire to have the shroud be the real burial cloth of Jesus. This unreasonable doubt is what drives the insinuations of unconscious bias having some kind of magical effect on objective machinery, a magical effect that even affects blind samples. Unreasonable doubt is where you ignore pieces of evidence which don't fit in with your preferred outcome. It's not scientific or rational, it's allowing your emotions to override your judgement. Laid out like that, can you see that your position is not one of reasonable doubt but of unreasonable doubt?
Nobody in this thread is trying to get you to abandon your faith (if you were to discuss your doubts about faith in another thread, that might be different). But faith in Jesus or God should not be affected by whether a piece of cloth is from the 1st or the 14th century, because nothing about the dating of the shroud has any bearing on whether Jesus existed or whether gods are real or imaginary. The evidence is what matters, and the evidence is that the shroud is from the 14th century.
The men of the Vatican don't allow the fact that the shroud is mediaeval to damage their faiths in their God, because rightly they understand that a piece of cloth doesn't have that power. Neither do most Christians because their faith is in Jesus and God, not a piece of cloth.
You've invested this piece of cloth with magical properties in the way it affects your faith. It's almost as if you have turned the shroud itself into a god. It isn't, it's just a piece of cloth which like hundreds of other religious objects has developed a raft of myths and superstitions around it. Just like all those fragments of the 'true cross' which, if put together would be enough to construct several hundred crosses and none of which came from an actual 1st century cross, let alone the actual cross on which Jesus was supposedly crucified.
I don't profess to fully understand the thinking which goes into the way you think about the shroud, but I would like you to consider this analogy about a hockey puck.
On my mantelpiece, I have an ice hockey puck which was used in a UK Challenge Cup final in 2002, being the last time the cup was won by the Sheffield Steelers (the team I support). I was at each leg of the final and have fantastic memories of the games.
If I one day discovered that the puck had been switched so that it's just an ordinary puck and not one used in that particular final, I wouldn't stop supporting the Sheffield Steelers, and I wouldn't feel that their victory had been tainted. I'd probably still keep the puck on the mantelpiece, because it's a reminder of the game, it's not the game itself or a substitute for my support for the team.
Just like that hockey puck, the only thing the shroud should be is a reminder of your faith just as any religiously-inspired painting, statue, piece of architecture or even piece of music. It can be that reminder of your faith no matter when it was produced, because it isn't the faith itself.