From the LWP::Simple manpage:

The user agent created by this module will identify itself as
"LWP::Simple/#.##" (where "#.##" is the libwww-perl version number) an
+d
will initialize its proxy defaults from the environment (by calling
$ua->env_proxy).
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I think they mean the HTTP_PROXY environment-variable, so in your script you could set something like:

$ENV{'HTTP_PROXY'}='http://proxy:3128/';
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before you create the user agent.