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.

---

---

Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!

Titles consisting of a single word are discouraged, and in most cases are disallowed outright.

Read Where should I post X? if you're not absolutely sure you're posting in the right place.

Please read these before you post! —

• How do I compose an effective node title?
• How do I post a question effectively?
• Markup in the Monastery

Posts may use any of the Perl Monks Approved HTML tags:

a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr

You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)

	For:		Use:
	&		&
	<		&lt;
	>		&gt;
	[		&#91;
	]		&#93;

Link using PerlMonks shortcuts! What shortcuts can I use for linking?

See Writeup Formatting Tips and other pages linked from there for more info.