I would try to process the log files in a single pass. You don't give a sufficient sample of your log files to be certain, but perhaps something like the following would work.

use strict;
use warnings;

my %transaction;

foreach my $file (<*.log>) {
    open(my $fh, '<', $file) or die "$file: $!";
    while(<$fh>) {
        chomp;
        if(/(.*?) \(([^\)]+)\)(.*)/) {
            my ($timestamp, $id, $log) = ($1, $2, $3);
            push(@{$transaction{$id}}, [ $timestamp, $id, $log, $file,
+ $. ]);
            my $status = status($transaction{$id});
            if($status eq 'complete') {
                submit($transaction{$id});
                delete($transaction{$id});
            } elsif($status eq 'corrupt') {
                report($status, $transaction{$id});
                delete($transaction{$id});
            }
        } else {
            warn "invalid input line: $_";
        }

    }
    close($fh);
}

foreach my $id (keys %transaction) {
    report('incomplete', $transaction{$id});
}
exit(0);



sub status {
    my $transaction = shift;

    return('incomplete') if(@$transaction < 3);
    return('corrupt') if(
        @$transaction > 3 or
        $transaction->[0][2] !~ /^(Authentication Request|Authenticati
+on Response|Accounting Request)/ or
        $transaction->[1][2] !~ /^Acct-Session-Id :/ or
        $transaction->[2][2] !~ /^User-Name :/
    );
    return('complete');
}

sub submit {
    my $transaction = shift;
    print "submit to database\n";
    foreach my $line (@$transaction) {
        my ($timestamp, $id, $log, $file, $line) = @$line;
        print "\t$file line $line: $timestamp ($id)$log\n";
    }

}

sub report {
    my ($status, $transaction) = @_;

    print STDERR "ERROR: $status transaction\n";
    foreach my $line (@$transaction) {
        my ($timestamp, $id, $log, $file, $line) = @$line;
        print STDERR "\t$file line $line: $timestamp ($id)$log\n";
    }
}
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