• Always use use strict;! It finds an error immediately in this case.
• s{ \A (\S+) \s+ (?= \d ) }{}xms only has one capture, so $2 will always be undef
• No line in File2 has a space followed by a digit, so s{ \A (\S+) \s+ (?= \d ) }{}xms will never match.

#!/usr/bin/perl

use strict;
use warnings;

my $qfn1 = "File1.txt";
my $qfn2 = "File2.txt";

my %positions;
{
   open(my $fh, '<', $qfn1)
      or die("Cannot open file \"$qfn1\": $!\n");

   while (<$fh>) {
      my ($key, $pos) = split /\s+/;
      $positions{$key} = $pos;
   }
}

{
   open(my $fh, '<', $qfn2)
      or die("Cannot open file \"$qfn2\": $!\n");

   for (;;) {
      defined( my $key = <$fh> )
         or last;
      defined( my $text = <$fh> )
         or last;
      chomp($key);
      chomp($text);

      defined( my $pos = $positions{$key} )
         or next;
      $pos = $pos - 1 - 3;
      $pos >= 0
         or next;
      my ($str) = $text =~ /^.{0,$pos}FAN(.{3})/
         or next;

      print("$key: $str\n");
   }
}
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12345: ABC
67890: JAK
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