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.
Carbon 14 testing is, by it's very nature, destructive. So the issue is, how much destruction is truly necessary?
My understanding is that over the years the amount of material needed has been greatly reduced. Is it today small enough so the Church could allow carefully carefully selected samples to be tested? I don't know, but would be most curious to learn the results.
Yes, I suspect the results would not be conclusive. But they might just rule out some possibilities, and that would be progress.
"Till such time, to portray the Church as putting obstacles in the way of useful research is slanderous."
Of course the Church puts "obstacles" in the way -- you have just listed some.
The question is whether those "obstacles" are reasonable and appropriate?
Naturally, you insist they are, and after the fiasco created by the previous carbon 14 tests, anyone can understand the Church's reluctance to rush into another one.
On the other hand, there comes a time when reason must overcome fear.
Imagine the operators of a carbon-14 testing lab -- would it not be the ultimate achievement to have met the Church's strict requirements and proposed a testing regime acceptable to all?