Serving dynamically cached images from nginx

I want to serve thumbnails for images, and these thumbnails are generated on demand, written to disk, then I want to hand them off to nginx to serve.

My root folder for thumbnails is /var/www/images. When I get a request for /thumb/post1/image1.jpg, I want to process it like this:

1. If the image is present in /var/www/images/thumb/post1/image1.jpg, serve it directly.
2. If the image is not there, it needs generating, so pass the request to the API at @backend.
3. The API generates the image and writes it to the thumbnail folder, then hands the path to it back to nginx using an X-Accel-Redirect header.
4. nginx restarts processing at step 1, which will succeed because the file is now present.
5. If the item the thumb was requested for doesn't exist, the API will return a 404, and nginx should serve a placeholder image located at /var/www/images/missing.png instead.

My nginx config looks like this:

upstream api {
  server localhost:7440 fail_timeout=0;
}

server {
  root /var/www/www.example.com/public;

  location / {
    try_files $uri @backend;
  }

  location /thumb {
    root /var/www/images;
    try_files $uri @backend /missing.png;
  }

  location @backend {
    root /var/www/api.example.com/public;
    proxy_set_header X-Real-IP $remote_addr;
    proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
    proxy_set_header X-Forwarded-Proto $scheme;
    proxy_set_header Host $http_host;
    proxy_set_header X-NginX-Proxy true;

    proxy_redirect off;
    proxy_pass http://api;

    #For websocket compatibility
    proxy_http_version 1.1;
    proxy_set_header Upgrade $http_upgrade;
    proxy_set_header Connection "upgrade";
  }
}

My thumbnails are kept outside the project folder, so I need a root directive in the /thumb location to make it look there for the image. The / location handles API requests and other static assets and, and the /thumb location does the same, but also has the fallback to missing.png.

One quirk: for historical reasons my overall root folder is not the same as the one used by my @backend named location, however, I'm overriding that in both location directives, and there is no top-level try_files.

However, this doesn't work. Requests for missing images don't get sent to the API, but a fallback for a missing image does! If I remove the fallback, requests do get sent to the API, but then the handoff to x-accel-redirect fails, even though the file now exists; when the thumb image does exist, nginx doesn't serve it – it just hits the API again.

How should this be configured?