I think that I would separate the concerns by prefiltering the files and feeding each thread's lines to different processes via pipes:

#! perl
use strict;

#### Warning! Untested code -- just an example.
for my $log ( <*.log> ) {
    open my $fhLog, '<', $log or die $!;

    my %kids;

    while( <$fhLog> ) {
        my( $tid ) = m[\(([0-9A-F]+)\)];
        if( not exists $kids{ $tid } ) {
            my $pid = open my $fh, '|-', 'perl non-interleaved.pl' or 
+die $!;
            $kids{ $tid } = [ $pid, $fh ];
        }
        print { $kids{ $tid }[ 1 ] } $_;
    }

    for my $kid ( keys %kids ) {
        close $kid->[ 1 ];           ## close the pipe
        waitpid $kid->[ 0 ], 0;      ##wait for the kid to finish
        rename $log, "archive/$log"; ## move the file
    }
}
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This way, you don't have to accumulate lots of lines in memory in order to get complete transactions. The same simplified non-interleaved.pl deals with each threads data, and deals with just one transaction at a time.