Is it possible to trace the password the application is attempting to use to connect?
The short answer is you can't see the password being used to connect.  That being said, you can restore a backup copy of the master database to a test instance, get the hashed password, and then apply it with the HASHED option of ALTER LOGIN.

The password hash can be retrieved from the restored master database with:

SELECT LOGINPROPERTY('your-sql-login', 'PasswordHash');


The password can then be applied to the other instance, substituting the value from the script above:

ALTER LOGIN your-sql-login
WITH PASSWORD = 0x01000CF35567C60BFB41EBDE4CF700A985A13D773D6B45B90900 HASHED;