Dynamic data to Cloudfront error pages

I have a cloudfront distribution setup to serve images from an s3 bucket, and a custom error page to handle 404 responses if the image does not exist in my bucket. The error page points to an API Gateway endpoint that generates an image given a set of parameters. Those parameters are provided by the client through the input URI when making a request to the cloudfront distribution.

Is there a way for me to pass that data to my API endpoint, like adding headers or query parameters to the url? The problem I am facing is that when the original client request is met with a 404 response, the error page handler makes a separate request to the desired endpoint (which is a static url), and does not seem to carry over any data from the original request.