You don't tell us where your approach fails and how.

Simply converting your curl command line to LWP::UserAgent using curl2lwp gives:

#!perl
use strict;
use warnings;
use WWW::Mechanize;
use HTTP::Request;


my $ua = WWW::Mechanize->new();
my $r  = HTTP::Request->new(
    'GET' => 'https://example.com/',
    [
        'Accept'        => '*/*',
        'Accept'        => 'application/json',
        'Authorization' => '<token>',
        'Host'          => 'example.com:443',
        'User-Agent'    => 'curl/7.55.1',
    ],

);
my $res = $ua->request( $r, );


__END__

Created from curl command line
curl -X GET "https://example.com" -H "accept: application/json" -H "au
+thorization:<token>"
[download]

If your question is how to extract the cookie from a browser request, have a look at the response you see in the browser.

If your question is how to extract the cookie automatically without involving a browser, first you need to find out what request you have to make to which URL, and also whether sending the username and password as a JSON string in a POST request is the correct approach there. If that is correct, then instead of inspecting the responseContent, inspect the response headers using $response->headers->as_string().

You already mention a variable suggestively named $cookies, but you never pass it anywhere. See the ->cookie_jar method of LWP::UserAgent on how to set a cookie jar for LWP::UserAgent.