I don't think you need to change the complete LWP. I don't even think you can, as SOAP::WSDL's client is a SOAP::Transport::HTTP::Client that comes with SOAP::Lite. This in turn is a subclass of LWP::UserAgent and inherits its methods, but it also adds more stuff which you need.
So what you need to do is set the client's cookie jar to the one you want. The interface of the client is still the one from LWP::UserAgent, so your HTTP::CookieJar::LWP should be fine.
You can get to the client with the $interface->get_transport()
method. Once you have that, you can use LWP::UserAgent's $ua->cookie_jar( $cookie_jar_obj )
method.
$interface->get_transport->cookie_jar(HTTP::CookieJar::LWP->new);
I believe that's all you need to do.