intermittent access denied from airbnb calendar on using node request - node.js

Question

I'm trying to write a very simple solution to download and parse a calendar file from my Airbnb. Airbnb provides the calendar in ical format, with a unique url for each user such as:
https://www.airbnb.com/calendar/ical/1234.ics?s=abcd
Where those numbers (1234/5678) are unique hex keys to provide some security.
Whenever I hit my (private) url it replies instantly with an ical if I'm using a browser. I can be using any browser, even one from a different country that has never visited airbnb.com before. (I've got remote access to a server I tried it from when debugging.)
In nodejs it works only about 10% of the time. Most of the time I get a 403 error with the text of You don't have permission to access (redacted url) on this server.
Example code:
const request = require('request');
request.get(url, (error, response, body) => {
if (!error && response.statusCode === 200) {
return callback(null, body);
}
return callback('error');
});
This is using the request package here: https://github.com/request/request
I've set it up in an async.whilst loop and it takes about 50 tries to pull down a success, if I set a multi-second delay between each one. (Btw, https://github.com/caolan/async is awesome, so check that if you haven't.)
If it failed EVERY time, that'd be different, but the fact that it fails only occasionally really has me stumped. Furthermore, browsers seem to succeed EVERY time as well.
curl [url] also works, every time. So is there something I'm not specifying in the request that I need to?
Edit 1:
As requested, more of the headers from the reply. I also thought it was rate-limiting me at first. The problem is this is all from the same dev-box. I can curl, or request from a browser without issue multiple times. I can come back in 24 hours and use the nodejs code and it'll fail the first time, or first 50 times.
headers:
{ server: 'AkamaiGHost',
'mime-version': '1.0',
'content-type': 'text/html',
'content-length': '307',
expires: 'Wed, 24 May 2017 17:23:28 GMT',
date: 'Wed, 24 May 2017 17:23:28 GMT',
connection: 'close',
'set-cookie': [ 'mdr_browser=desktop; expires=Wed, 24-May-2017 19:23:28 GMT; path=/; domain=.airbnb.com' ] },
rawHeaders:
[ 'Server',
'AkamaiGHost',
'Mime-Version',
'1.0',
'Content-Type',
'text/html',
'Content-Length',
'307',
'Expires',
'Wed, 24 May 2017 17:23:28 GMT',
'Date',
'Wed, 24 May 2017 17:23:28 GMT',
'Connection',
'close',
'Set-Cookie',
'mdr_browser=desktop; expires=Wed, 24-May-2017 19:23:28 GMT; path=/; domain=.airbnb.com' ],
trailers: {},
rawTrailers: [],
upgrade: false,
url: '',
method: null,
statusCode: 403,
statusMessage: 'Forbidden',