However, I am not sure how to apply a substitution, s///, to an element containing more that just text.

The documentation of HTML::Element's content_refs_list gives an example of how to modify text nodes contained in an element and the documentation of HTML::Element::traverse shows how to use a recursive function to walk the tree. Putting those together:

sub html_trim {
    my $elem = shift;
    for my $itemref ($elem->content_refs_list) {
        if ( ref $$itemref )
            { html_trim($$itemref) } # remove this for non-recursive
        else { $$itemref =~ s/^\s+|\s+$//g }
    }
}
for my $elem ($xhtml->findnodes('//div/ul/li')) {
    html_trim($elem)
}
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