When I use LWP, I use the post() method instead of directly constructing the HTTP::Request object myself, and have never run into any issues. The Content for your request should be the user and pass arguments from your URL -- which, as a side effect, keeps the web server from logging user account info into its access log. I suspect that's the reason you wanted to POST in the first place.
In any case, this would give you something like the following:
my $auser = 'jebediah'; # kerman
my $apwd = 'KSP#bada55'; # green
my $page = 'http://api.xxx.com/endpoint';
my @args = ( user => $auser, pass => $apwd );
my $agent = new LWP::UserAgent();
die "couldn't create agent\n" unless $agent;
my $response = $agent->post( $page, Content => \@args );
die "couldn't post web page\n" unless $response;
# process $response->header() here if desired.
print $response->content(), "\n";
UPDATE: cleaned up the account info so i'm not hard-coding @args as much.