I personally would opt for the most straight forward way: Create a sub that logs the results to a file, and then call die.

    # excuse the sloppy code...
    sub log_and_die
    {
          my $msg = shift;
          # suppose $fh is an already defined, opened
          # file handle. of course, you can open one here
          # as well...
          print $fh $msg;
          die $msg;
    }

    DBI->connect( ... ) || log_and_die( $! )
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