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With the history of partizan bias displayed by the testrers so far, I would hope that the Church does not allow every nick dick and harry to “test” the Shroud. I have much greater trust in the Church than in the habitual liars in the scientific community. Just look at the climatologists. I would not buy a used car from these shysters.

The chain of custody is obviously not clear, but the fact that the techological process that had produced the shroud has not been identified, and every indication is that it is from pre-medieval Palestine and displays the pathologies of the death by flogging and crucifixion, and burial according to the Jewish custom of the time convince me.

That another carbon dating will show some spectrum outside of 1 c would not prove anything, on the other hand, precisely because the chain of custody is not known. The first time they were looking at soot from a fire that occurred in the Middle Ages. Next time it will be something else like that. It is not terribly relevant what carbon dating will find.

"I would hope that the Church does not allow every nick dick and harry to “test” the Shroud."

Of course, there are relatively simple solutions to the problems you raise. So let me suggest just a few.

The Church has any number of ancient articles for which the dates are well known. And there are many labs around the world capable of doing state-of-the-art carbon 14 testing. So let the Church submit known articles to various labs, and see which ones produce the most accurate results.
Your charge that ALL scientists are just a bunch of lying athiests is not 100% true. Many scientists are devout Christians, indeed devout Catholics. So it would be a simple matter for the Church to insist that recognized Catholic scientists are allowed to review every single step in the process of carbon 14 dating.
Finally, we have to recognize that various tests are going to produce different results. The prime example of this so far is the Shroud repair work selected and dated to the middle ages. Other sections of the Shroud could well yield other dates, and possibly no part may yield a 1st century date, all for reasons we may not yet understand. Nevertheless, these dates would still tell us a great deal about what has happened to the Shroud over millenia. And no doubt lead to other interesting questions.