### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

IaC = CDK.

Attempting to connect an API-GW endpoint directly to EventBridge.

```
      const awsIntegration = new apigateway.AwsIntegration({
      service: 'events',
      action: 'PutEvents',
      integrationHttpMethod: 'POST',
      options: {
        credentialsRole: role,
        requestTemplates: {
          'application/json': requestTemplate,
        },
        passthroughBehavior: apigateway.PassthroughBehavior.NEVER,
        integrationResponses: [
          successResponse,
          serverErrorResponse,
          clientErrorResponse,
        ],
      },
    });
```

Receiving response:

```
    {
      statusCode: 400,
      data: {
        Type: 'User',
        message: `Error invoking integration for API Gateway ID '1z7ky5zzcb': API Gateway AWS integration action URI "arn:aws:apigateway:ap-southeast-2:events:action/PutEvents", method "GET" not yet implemented`,
        __type: 'InvalidRequest'
      }
    }
```

Similar to bug reported:

* https://github.com/localstack/localstack/issues/5508

### Expected Behavior

* 200 response
* Event created in relevant Eventbus

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

docker run localstack/localstack

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

Calling the API endpoint using axios from jest tests.

### Environment

```markdown
- OS: MacOS
- LocalStack: latest
```


### Anything else?

_No response_
uri = integration.get("uri") or integration.get("integrationUri") or ""
        s3 = aws_stack.connect_to_service("s3")
        uri = apply_request_parameters(
            uri,
            integration=integration,
            path_params=path_params,
            query_params=query_string_params,
        )
        uri_match = re.match(self.TARGET_REGEX_PATH_S3_URI, uri) or re.match(
            self.TARGET_REGEX_ACTION_S3_URI, uri
        )
        if not uri_match:
            msg = "Request URI does not match s3 specifications"
            LOG.warning(msg)
            return make_error_response(msg, 400)
        bucket, object_key = uri_match.group("bucket", "object")
        LOG.debug("Getting request for bucket %s object %s", bucket, object_key)
        try:
            object = s3.get_object(Bucket=bucket, Key=object_key)
        except s3.exceptions.NoSuchKey:
            msg = f"Object {object_key} not found"
            LOG.debug(msg)
            return make_error_response(msg, 404)
        headers = aws_stack.mock_aws_request_headers(service="s3")
        if object.get("ContentType"):
            headers["Content-Type"] = object["ContentType"]
        
        response = request_response_stream(stream=object["Body"], headers=headers)
        return response
class HTTPIntegration(BackendIntegration):