API Gateway Service Integration to Data Firehose

I am trying to setup an API Gateway proxy service integration to Data Firehose. (Importantly, this has nothing to do with streaming access logs to Firehose)

However, I cannot seem to get it to work. Can anyone point me to an example?

Currently the integration responds with a 200, but Firehose never processes any records. So I don't know if Firehose actually receives the event or if I have an error in my proxy definition (below) which forces the 200.

Any support appreciated, I've been banging my head against the wall over this...

Here is what I have for my integration right now:

openapi: "3.0.1"
servers:
- url: "https://my-api-host.com"
paths:
  /v1/events:
    post:
      requestBody:
        content:
          application/json:
            schema:
              $ref: "#/components/schemas/ApiV1AnalyticsFlowEventsRequest"
        required: true
      responses:
        "400":
          description: "400 response"
          headers:
            Access-Control-Allow-Origin:
              schema:
                type: "string"
            Access-Control-Allow-Methods:
              schema:
                type: "string"
            Access-Control-Allow-Headers:
              schema:
                type: "string"
          content:
            application/json:
              schema:
                $ref: "#/components/schemas/ApiV1AnalyticsFlowEventsErrorResponse"
        "200":
          description: "200 response"
          headers:
            Access-Control-Allow-Origin:
              schema:
                type: "string"
            Access-Control-Allow-Methods:
              schema:
                type: "string"
            Access-Control-Allow-Headers:
              schema:
                type: "string"
          content:
            application/json:
              schema:
                $ref: "#/components/schemas/ApiV1AnalyticsFlowEventsErrorResponse"
      x-amazon-apigateway-integration:
        type: "aws"
        credentials: "arn:aws:iam::1234567890:role/flow-events-firehose-integration"
        httpMethod: "POST"
        uri: "arn:aws:apigateway:us-west-2:firehose:action/PutRecordBatch"
        responses:
          "4\\d{2}":
            statusCode: "400"
            responseParameters:
              method.response.header.Access-Control-Allow-Methods: "'DELETE,GET,HEAD,OPTIONS,PATCH,POST,PUT'"
              method.response.header.Access-Control-Allow-Headers: "'Content-Type,Authorization,X-Amz-Date,X-Api-Key,X-Amz-Security-Token'"
              method.response.header.Access-Control-Allow-Origin: "'*'"
            responseTemplates:
              application/json: "\n#set($data = $input.path('$'))\n{\n  \"data\":\
                \ \"$util.base64Encode($data)\"\n}"
          "2\\d{2}":
            statusCode: "200"
            responseParameters:
              method.response.header.Access-Control-Allow-Methods: "'DELETE,GET,HEAD,OPTIONS,PATCH,POST,PUT'"
              method.response.header.Access-Control-Allow-Headers: "'Content-Type,Authorization,X-Amz-Date,X-Api-Key,X-Amz-Security-Token'"
              method.response.header.Access-Control-Allow-Origin: "'*'"
            responseTemplates:
              application/json: "\n#set($failed = $input.path('$.FailedPutCount'))\n\
                #if($failed > 0)\n{\n  \"data\": \"$util.base64Encode($data)\"\n}\n\
                #else\n{}\n#end\n"
        requestTemplates:
          application/json: "\n#set($data = $input.path('$.events'))\n{\n  \"DeliveryStreamName\"\
            : \"flow-events-ingestion\",\n  \"Records\": [\n    #foreach($elem in\
            \ $data)\n      {\n        \"Data\": \"$util.base64Encode($elem)\"\n \
            \     }#if($foreach.hasNext),#end\n    #end\n  ]\n}"
        passthroughBehavior: "when_no_match"
components:
  schemas:
    ApiV1AnalyticsFlowEventsRequest:
      required:
      - "events"
      type: "object"
      properties:
        events:
          type: "array"
          items:
            type: "string"
      additionalProperties: false
    ApiV1AnalyticsFlowEventsErrorResponse:
      type: "object"
      properties:
        data:
          type: "string"
      additionalProperties: false

The referenced role looks like:

{
    "Version": "2012-10-17",
    "Statement": [
        {
            "Effect": "Allow",
            "Principal": {
                "Service": "apigateway.amazonaws.com"
            },
            "Action": "sts:AssumeRole"
        }
    ]
}

{
    "Version": "2012-10-17",
    "Statement": [
        {
            "Action": "firehose:PutRecordBatch",
            "Resource": "arn:aws:firehose:*:1234567890:deliverystream/flow-events-ingestion",
            "Effect": "Allow"
        }
    ]
}