### 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`
queue_arn_1 = sqs_queue_arn(queue_url_1)
            mapping_uuid = lambda_client.create_event_source_mapping(
                EventSourceArn=queue_arn_1,
                FunctionName=function_name,
                MaximumBatchingWindowInSeconds=1,
            )["UUID"]
            _await_event_source_mapping_enabled(lambda_client, mapping_uuid)
            sqs_client.send_message(QueueUrl=queue_url_1, MessageBody=json.dumps({"foo": "bar"}))
            retry(
                check_expected_lambda_log_events_length,
                retries=10,
                sleep=1,
                function_name=function_name,
                expected_length=1,
                logs_client=logs_client,
            )
            rs = sqs_client.receive_message(QueueUrl=queue_url_1)
            assert rs.get("Messages") is None
        finally:
            lambda_client.delete_event_source_mapping(UUID=mapping_uuid)
class TestDynamoDBEventSourceMapping:
    def test_dynamodb_event_source_mapping(
        self,
        lambda_client,
        create_lambda_function,
        create_iam_role_with_policy,
        dynamodb_client,
        dynamodb_create_table,
        logs_client,
        check_lambda_logs,