An easy way to get a start is by converting your Curl command to a Perl program using (my) Curl to Perl converter:

#!perl
use strict;
use warnings;

use LWP::UserAgent;

my $ua = LWP::UserAgent->new( 'send_te' => '0' );
my $r  = HTTP::Request->new(
    'GET' => 'https://example.com/',
    [
        'Accept'     => '*/*',
        'User-Agent' => 'curl/7.55.1',
        'Cookie'     => 'session=536...035a'
    ],

);
my $res = $ua->request( $r, ':content_file' => 'HTML_output.txt' );



__END__

Created from curl command line
curl https://example.com --cookie "session=536...035a" -o "HTML_output
+.txt"
[download]

I'm not sure if your Firefox stores its cookies in a cookies.txt file still or if it stores them in an SQLite database nowadays.