I was having trouble with LWP & proxies until I found that it was better to set proxy support using the environmental variable method. The Crypt::SSLeay docs says:

The CONNECT method is used by Crypt::SSLeay's internal proxy support.

Definately read that whole document.

So what you want is something like this:

$ENV{HTTP_PROXY} = 'http://192.168.250.11:8118/'; 
$ENV{HTTPS_PROXY} = 'http://192.168.250.11:8118/';
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