It's super easy to find the desired LI using XPath.

use HTML::TreeBuilder        qw( );
use HTML::TreeBuilder::XPath qw( );

my $html = do{local $/; <DATA>};
my $tree = HTML::TreeBuilder->new_from_content($html);

my @results = $tree->findnodes(
   '//li/*[1][name()="span"]/@class/parent::*/parent::*'
);
foreach my $li (@results) {
   # $li holds the HTML::Element object for the LI.
   ...
}

$tree->delete();
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It's a little bit tricky to serialize the LI without the LI start tag, the SPAN element and the LI end tag.

use HTML::Entities qw( encode_entities );

sub node_as_html {
   my ($node) = @_;
   if (ref($node)) {
      my $html = $node->as_HTML(undef, undef, {});
      chomp($html);
      return $html;
   } else {
      return encode_entities($node);
   }
}

...
   my @children = $li->content_list();
   shift(@children);  # Skip SPAN
   print(node_as_html($_))
      foreach @children;
   print("\n");
...
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