EventBridge API Destination Not Sending Origin Header In Post Request

We have a business requirement to transform and send a request via a POST, with a custom 'origin' header value.

When trying to send the request with the header key 'origin' and the header value 'blah' I can see that it arrives at the destination with any origin header. If I send the header 'origin1' with value blah, I can see this arrive at the destination.

Is there a list of hidden reserve headers that cannot be modified?

Is an alternative solution to integrate with API Gateway or a Lambda that could add this header and send the request on?