### 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_
resp = aws_client.sqs.receive_message(QueueUrl=queue_url)
            return resp["Messages"]
        messages = retry(get_message, retries=3, sleep=1, queue_url=queue_url)
        assert len(messages) == 1
        actual_event = json.loads(messages[0]["Body"])
        self.assert_valid_event(actual_event)
        assert actual_event["detail"] == EVENT_DETAIL
        
        self.cleanup(bus_name_1, rule_name, target_id)
        self.cleanup(bus_name_2)
        aws_client.sqs.delete_queue(QueueUrl=queue_url)
    def test_put_events_with_target_lambda(self, create_lambda_function, cleanups, aws_client):
        rule_name = f"rule-{short_uid()}"
        function_name = f"lambda-func-{short_uid()}"
        target_id = f"target-{short_uid()}"
        bus_name = f"bus-{short_uid()}"
        
        cleanups.append(lambda: aws_client.awslambda.delete_function(FunctionName=function_name))
        cleanups.append(lambda: self.cleanup(bus_name, rule_name, target_id))
        rs = create_lambda_function(
            handler_file=TEST_LAMBDA_PYTHON_ECHO,
            func_name=function_name,
            runtime=Runtime.python3_9,
        )
        func_arn = rs["CreateFunctionResponse"]["FunctionArn"]
        aws_client.events.create_event_bus(Name=bus_name)
        aws_client.events.put_rule(
            Name=rule_name,