API Gateway as Reverse HTTP Proxy to SQS

I am trying to use AWS API Gateway as a reverse (forwarding) proxy to AWS SQS. I essentially want to send a REST request to the API Gateway which then gets forwarded directly to the SQS REST API and returns the response.

When I send a request to the gateway, I immediately get back

<?xml version="1.0"?>
  <ErrorResponse xmlns="http://queue.amazonaws.com/doc/2012-11-05/">
    <Error>
      <Type>Sender</Type>
      <Code>AccessDenied</Code>
      <Message>Access to the resource https://sqs.us-east-1.amazonaws.com/ is denied.</Message>
      <Detail/>
    </Error>
    <RequestId>51c903b2-4da3-5d5e-a3b8-589ee72167de</RequestId>
  </ErrorResponse>

However, when I switch the request URL to SQS directly (https://sqs.us-east-1.amazonaws.com) the request succeeds.

What am I missing?

curl --request POST 'https://my-api-gateway.com/sqs' \
--header 'X-Amz-Date: <date>' \
--header 'X-Amz-Security-Token: <token>' \
--header 'Authorization: <auth>' \
--header 'Amz-Sdk-Invocation-Id: <invocation>' \
--header 'Amz-Sdk-Request: attempt=1; max=10' \
--header 'User-Agent: aws-sdk-go-v2/1.16.5 os/macos lang/go/1.18.3 md/GOOS/darwin md/GOARCH/arm64 api/sqs/1.18.6' \
--header 'Content-Length: 206' \
--data-urlencode 'Action=ReceiveMessage' \
--data-urlencode 'MaxNumberOfMessages=10' \
--data-urlencode 'QueueUrl=<my-queue-url>' \
--data-urlencode 'Version=2012-11-05' \
--data-urlencode 'WaitTimeSeconds=20'

Configuration:

1. Integrations
2. Routes
3. Parameter Mappings