I can unequivocally say that what choroba has there works properly. Here's an actual working example I just tested (with some name/data modifications):

use warnings;
use strict;

use LWP::UserAgent;
use JSON;

my $ua = LWP::UserAgent->new;

my $body_content = {
    token => 'NTE2M2Q3YmFmMGEzNjEzMGQyZmFmYmViNGEasdfmNzkzMTE3NzI3MWMz
+MDhkZjliYWE3NTI1OTg1MDQwNjNlZg'
};

my $resp = $ua->post(
    'https://my.server.com',
    'Content-Type'    => 'application/json',
    'Content'         => encode_json $body_content
);

print $resp->decoded_content;
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