Triggers and the Transaction
Probably not.  Violating a declarative constraint will always raise an error and prevent the transaction from committing writes that would violate the constraints.  The point of database constraints - checks, foreign keys, uniques etc. - is that they prevent invalid data from being recorded in the database by aborting writes that will violate the constraints.  The constraint violation raises an error and rolls back the transaction.

If you want the transaction to fail gracefully you will either have to deal with the exception or pre-validate the data before attempting to write it.  In the latter case you will have to replace the validations in the stored procedures or implement validation in the application.