How to pass API Gateway authorizer context to a HTTP integration

I have successfully implemented a Lambda authorizer for my AWS API Gateway, but I want to pass a few custom properties from it to my Node.js endpoint.

My output from my authorizer follows the format specified by AWS, as seen below.

{
  "principalId": "yyyyyyyy",
  "policyDocument": {
    "Version": "2012-10-17",
    "Statement": [
      {
        "Action": "execute-api:Invoke",
        "Effect": "Allow|Deny",
        "Resource": "arn:aws:execute-api:<regionId>:<accountId>:<appId>/<stage>/<httpVerb>/[<resource>/<httpVerb>/[...]]"
      }
    ]
  },
  "context": {
    "company_id": "123",
    ...
  }
}

In my case, context contains a few parameters, like company_id, that I would like to pass along to my Node endpoint.

If I was to use a Lambda endpoint, I understand that this is done with Mapping Template and something like this:

{
  "company_id": "$context.authorizer.company_id"
}

However, Body Mapping Template is only available under Integration Request if Lambda is selected as Integration type. Not if HTTP is selected.

In short, how do I pass company_id from my Lambda authorizer to my Node API?