SOAP::Lite implements the server part through https://metacpan.org/pod/distribution/SOAP-Lite/lib/SOAP/Transport.pod#SOAP::Transport::HTTP and https://metacpan.org/pod/distribution/SOAP-Lite/lib/SOAP/Transport.pod#SOAP::Transport::HTTP::Server.

From reading the documentation, you can either replace the server class by your own server or monkeypatch the ->response method to modify the response when it gets stored.

use SOAP::Transport::HTTP

my $old_response = \&SOAP::Transport::HTTP::Server::response;
*SOAP::Transport::HTTP::Server::response = sub {
    my $resp = $old_response->( @_ );
    $resp->headers->header('Access-Control-Allow-Origin' => '*'); # is
+ this wise?!
    return $resp;
};
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