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"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.

See the factual reply by Swordmaker just above.

In principle, of course, there is no objection to non-destructive testing with mature technology by honest people, and at such time that these three conditions obtain, I am sure the Church will agree to testing.

Till such time, to portray the Church as putting obstacles in the way of useful research is slanderous.