#!/usr/bin/perl
use warnings;
use strict;
use Fcntl qw/:DEFAULT :flock/;

my $fileA = '/path/to/file_a.txt';
my $fileB = '/path/to/file_b.txt';
my $fileC = '/path/to/file_a.bak';

my $tagline = qr/\QFile A: *** Initial Temperature/;

sysopen my $fc, $fileC, O_WRONLY | O_EXCL
    or die 'Competing process: ', $!;
flock $fc, LOCK_EX | LOCK_NB
    or die 'Competing process: ', $!;

open my $fa, '+<', $fileA or die $!;
flock $fa, LOCK_EX;

while (<$fa>) {
    print $fc $_;
    last if /^$tagline/;
}

open my $fb, '<', $fileB or unlink $fileC and die $!;
print $fc $_ while <$fb>;
close $fb or unlink $fileC and die $!;

print $fc $_ while (<$fa>);
close $fc or unlink $fileC and die $!;

# reverse the order of the next two statements if
# your system won't clobber the open file
rename $fileC, $fileA;
close $fa;