...The methods inherited from HTML::Parser are used forbuilding the HTML tree, and the methods inherited from HTML::Element are what you use to scrutinize the tree. Besides this (HTML::TreeBuilder) documentation, you must also carefully read the HTML::Element documentation, and also skim the HTML::Parser documentation -- probably only its parse and parse_file methods are of interest.

Loosely put (in the context of your question and of HTML::Element and HTML::Parser), since <br> is NOT "as_text" - it's html - neither it nor any substitute appears the output:

$h->as_text(skip_dels => 1)
Returns a string consisting of only the text parts of the element's descendants. (Emphasis supplied)

...and NOT just BTW,
you've offered no code and your narrative cites a different module (HTML::TreeBuilder) than your title (HTML::Element). Those shortcomings make offering help more difficult than necessary.

Edited: removed irrelevant detail from the second quote from the docs; emphasized the relevant part.

#!/usr/bin/perl

use strict;
use warnings;
require HTML::TreeBuilder;

my $file = <DATA>;
my $tree = HTML::TreeBuilder->new->parse_file($file);

require HTML::FormatText;

my $formatter = HTML::FormatText->format_string(
                $file,
                leftmargin => 3,
                rightmargin => 72);
print $formatter;

__DATA__
<p>The Cow Jumped<br>Over The Moon</p>
[download]

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);
[download]

It’s true:   “tags” are removed, and <br&rt; is a tag.

When faced with this problem... I cheated.   I used this regex: s/<br>/ /g ... which turns all of those tags into one blank space.

“Well, on the one hand, it’s a butt-ugly hack that you would never want to tell your mother about.   But, on the other hand ... it works.”

Oh, fudge.   Perlmonks logged me out.   You probably have to backslash-escape the left and right angle brackets.   (But you already knew that, didn’t you?)