I am not sure where you get that the Shroud of Turin may have been repaired (at the corner tested) as being "contamination and conspiracy".
The Shroud of Turin was in a fire and it was held by the edges. So the fact it was repaired, and that is documented in the history, should not be a surprise.
The original protocol required samples tested from different locations (by the scientists).
The church only allowed the "edge" of the Shroud to be "torn off" for the tests. This is the fault of the church.
A better way to resolve the "issues" around C-14 dating is to do "microscopic inspection" of the fabric of any material used for C-14 dating to make sure it is not done with fabric "interwoven" as part of a repair.