As pointed out in https://render.discourse.group/t/maintenance-mode-feedback/39463, it's not possible to use Maintenance Mode through JSON.

If back-end and front-end are separate, it's not possible to properly communicate with back-end in maintenance mode from the client, due to the CORS issue.

So instead of getting a proper 503 error status back, the client just hits a CORS error. If Render could add CORS support to the maintenance mode, this issue would go away. I wouldn't think it's a huge undertaking, so it would be great to have it looked at.