There are two Ardabil carpets, one in the Victoria & Albert Museum in
London, the other in the Los Angeles County Museum of Art.  They are
dated 1539/1540.  What method(s) were used for verification?  When
were they authenticated?  Have further tests been done since?  Or was
the date in the woven cartouche at one end of the carpet used as the
only proof?

In my experience, this is usually done by people who know about the
styles of the time/place in question.  Of course, carbon dating is
possible.  It would be accurate to within a few years.  I imagine, if
it was really important, both methods would be used.  1500 is not that
long ago, and lots of records would still be around.