You should compare what curl sends with what you send.

Invoke curl with -v -v -v to see what it sends over the wire.

Look at that.

Look at your script and the parameters you give it, and look at the API documentation for the API.

When I run your code through my Curl to Perl converter, I get the following:

#!perl
use strict;
use warnings;

use HTTP::Tiny;

my $ua  = HTTP::Tiny->new( 'verify_SSL' => '1' );
my $res = $ua->request(
    'POST' => 'https://api.link.io/v1/payment',
    {
        headers => {
            'Accept'         => '*/*',
            'User-Agent'     => 'curl/7.55.1',
            'Content-Length' => '196',
            'Content-Type'   => 'application/json',
            'X-Api-Key'      => '{{api-key}}'
        },
        content =>
"{  \x22price_amount\x22: 3999.5,  \x22price_currency\x22: \x22usd\x22
+,  \x22pay_currency\x22: \x22usd\x22,  \x22ipn_callback_url\x22: \x22
+https://link.io\x22,  \x22order_id\x22: \x22RGDBP-21314\x22,  \x22ord
+er_description\x22: \x22Apple Macbook Pro 2019 x 1\x22}"
    },
);
[download]

Most likely, the API does not want {{ABCDEF}} but ABCDEF in its x-api-key header.