What about this? I keep a list of possible positions, I delete non-matching positions recursively character by character. If I make the string much longer (x 100_000), it still runs only for 8 seconds on my box.

#!/usr/bin/perl
use warnings;
use strict;
use feature qw(say);

#                         1         2         3         4
#               0123456789012345678901234567890123456789012345
#               |         |         |    |   |        |
our $string  = '1234561234123x561234123x61234x3456123412345x1';
our $pattern = '123456';

our $pattern_length = length $pattern;

our %results;

search(0, 0, [0 .. length($string) - 1]);
say for sort { $a <=> $b } keys %results;

sub search {
    my ($p_pos, $s_pos, $positions) = @_;
    return if $p_pos > $pattern_length;
    return unless @$positions;

    if ($p_pos == $pattern_length) {
        undef @results{@$positions};
        return;
    }

    my $char = substr $pattern, $p_pos, 1;

    search($p_pos + 1, $s_pos + 1,
           [ grep {
               my $ch = substr($string, $s_pos + $_, 1);
               $char eq $ch or 'x' eq $ch;
           } @$positions ]);

    search($p_pos + 2, $s_pos + 1,
           [ grep substr($string, $s_pos + $_, 1) eq 'x',
                  @$positions ]);
}
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