To destroy the ua object, you could throw the new() call in the loop. I don't know if that will prevent caching, because of the proxy, but you could try this.

use LWP::UserAgent;
use Time::HiRes 'time', 'sleep';

$count = 5;

while ($count--)
{
    my $ua = LWP::UserAgent->new;
    $ua->proxy ('http', 'http://10.12.8.12:8080');
    $start = time();
    $response = $ua->get("http://myaccount.polymerchemicals.com/ping.h
+tm");
    $latency = time() - $start;
    print "Received in $latency seconds\n";
    sleep (60);    
}
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