If $ua->env_proxy; doesn't work, here is another example I have used in the past.

sub GetURL {
    ##########
    #
    # GetURL is a generic LWP sub used to GET a URL using the provided
    #         proxy settings, upon success it returns the page content
+s or UNDEF
    #
    ##########

    my $url = shift;
    my $userAgent = new LWP::UserAgent;
    $userAgent->timeout( 3 ); # 3 sec timeout
    $userAgent->proxy( 'http' , $PROXY ) if $PROXY;
    my $request = HTTP::Request->new( GET => "$url" );
    $request->proxy_authorization_basic( $USERNAME , $PASSWORD ) if $P
+ASSWORD;
    my $result = $userAgent->request( $request) ;
    if ( $result->is_success ) {
        return $result->content;
    }
    return undef;
}
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