I found some code that does just that in a script I use. It uses the XML::Twig interface, for I prefer that to HTML::Tree. Here's the subroutine, plus some code to show how it should be invoked. Of course, you may need to also teach this about some elements other than br.

use warnings; use strict; use 5.010;
use XML::Twig;
sub html_text {
        my($el) = @_;
        my $r = "";
        for my $n ($el->descendants_or_self) {
                if ($n->is_text) {
                        $r .= $n->trimmed_text;
                } elsif ("br" eq $n->gi) {
                        $r .= "\n";
                }
        }
        $r;
}
my $tw = XML::Twig->new;
$tw->parse_html(q(<p>The Cow Jumped<br>Over the Moon</p>));
say html_text($tw->root);
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