Thanks to all for a lot of invaluable knowledge.

In the end, I applied a variation on dws' sliding window technique in "Match in Huge Files." It's proven to completely eliminate the memory usage issue.

The solution (a.k.a. 'The Way Tekkie Did It'):

The addition of the following search subroutine:

sub search {

    my ($file, $searchString) = @_;

    local *F;
    open(F, "<", $file) or return 0;
    binmode(F);

    my $nbytes = read(F, my $window, 2 * BLOCKSIZE);

    while ( $nbytes > 0 ) {

        if ( $window =~ m/$searchString/ ) { close(F); return 1; }
        
        $nbytes = read(F, my $block, BLOCKSIZE);
        last if $nbytes == 0;

        substr($window, 0, BLOCKSIZE) = '';
        $window .= $block;
    }

    close(F);

    return 0;
}
[download]

Coupled with a modification to the main loop here:

if(search($file, $searchString)) { 

    print "\tFound: $file\n";

        foreach(@dir) { 

            if(/^.*?\s+([A-Z]{1}$numSet\.\S{3})$/) {

            my $found = join('\\', ($dirPath, $1));
            print "\tFound: $found\n"; 
        }
    }
}
[download]

And you've got another problem solved by the wisdom of the monks.
Many thanks once more to all who offered their assistance.