The deletion of <br> is no different than the deletion of the paragraph tags. Even the doc for the module (HTML::TreeBuilder) mentioned in your description of your problem (See ' ...and NOT just BTW') tells you quite plainly where to find the answer to this question:

...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.