### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

I Created an REST APIGatway with lambda handler using AWS CDK. when I hit the API the response header content type is always  set to "text/html"  instead of `application/json` . even if I changed the response template or added the header to lambda returned response. when I deploy the same stack on AWS account  the response header is fine and work as expected

### Expected Behavior

Response Header content-type to be `application/json`

### How are you starting LocalStack?

  `localstack start` command

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command`)

    `localstack start` command

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

I'm using AWS CDK typescript application and deploy it using `cdklocal` command on localstack

Using CDK follow the steps:
- create  REST ApiGateway
- create Lambda function 
- integrate lambda with APIGatway

On Command line:
- deploy using `cdklocal deploy` command

Finally
- test the endpoint and check the response header content-type




### Environment

```markdown
- OS: Windows 
- LocalStack: latest
```


### Anything else?

When I deploy the same stack on AWS account everything works fine and the response header content-type is `application/json`
rs = lambda_client.create_event_source_mapping(
                    EventSourceArn=queue_arn_2,
                    FunctionName=function_name,
                    BatchSize=BATCH_SIZE_RANGES["sqs"][1] + 1,
                )
            snapshot.match("invalid-create-event-source-mapping", e.value.response)
            e.match(INVALID_PARAMETER_VALUE_EXCEPTION)
        finally:
            lambda_client.delete_event_source_mapping(UUID=uuid)
    def test_sqs_event_source_mapping(
        self,
        create_lambda_function,
        lambda_client,
        sqs_client,
        sqs_create_queue,
        sqs_queue_arn,
        logs_client,
        lambda_su_role,
    ):
        function_name = f"lambda_func-{short_uid()}"
        queue_name_1 = f"queue-{short_uid()}-1"
        try:
            create_lambda_function(
                func_name=function_name,
                handler_file=TEST_LAMBDA_PYTHON_ECHO,
                runtime=LAMBDA_RUNTIME_PYTHON36,
                role=lambda_su_role,
            )
            queue_url_1 = sqs_create_queue(QueueName=queue_name_1)