Send QR code via SMS?
As mentioned in my comment to Demerit's answer I successfully achieved my objective to send a QR code to an event participant's phone (albeit the URL and not the image).

I installed the Clickatell extension for CiviSMS and the QR Code Checkin extension to generate the emails or SMSs to send the QR Code or its URL to event participants. 

The instructions to set up the Clickatell extension at https://docs.civicrm.org/user/en/latest/sms-text-messaging/set-up/ are outdated but it was not hard to set it up.

The QR code extension was easy to set up, but don't forget to manually set the required permissions in your CRM (Drupal in my case). The extension provides a handy couple of tokens that can be easily included in an email or SMS.

The downside of the QR Code extension is that it only works on one event at a time. If you have multiple events that run on different days it may be possible to change the 'effective' event (on the event page) before sending QR codes and again during checkin.  I haven't tested this.

Below is an example of the test SMS I generated.