You are creating $ua, but LWP:Simple already has a $ua. You need to import $ua instead of creating your own, since yours will not get used by LWP::Simple.

use LWP::Simple qw( $ua getstore );
# ...
$ua->cookie_jar(
  HTTP::Cookies->new( file => $cookie_path, autosave => 1 )
);
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This is documented in LWP::Simple, but only by a single line of text, with no example.