### 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`
runtime=LAMBDA_RUNTIME_PYTHON37,
                role=role_arn,
            )
            kinesis_client.create_stream(StreamName=kinesis_name, ShardCount=1)
            result = kinesis_client.describe_stream(StreamName=kinesis_name)["StreamDescription"]
            kinesis_arn = result["StreamARN"]
            wait_for_stream_ready(stream_name=kinesis_name)
            queue_event_source_mapping = sqs_create_queue()
            destination_queue = sqs_queue_arn(queue_event_source_mapping)
            destination_config = {"OnFailure": {"Destination": destination_queue}}
            message = {
                "input": "hello",
                "value": "world",
                lambda_integration.MSG_BODY_RAISE_ERROR_FLAG: 1,
            }
            result = lambda_client.create_event_source_mapping(
                FunctionName=function_name,
                BatchSize=1,
                StartingPosition="LATEST",
                EventSourceArn=kinesis_arn,
                MaximumBatchingWindowInSeconds=1,
                MaximumRetryAttempts=1,
                DestinationConfig=destination_config,
            )
            event_source_mapping_uuid = result["UUID"]
            _await_event_source_mapping_enabled(lambda_client, event_source_mapping_uuid)
            kinesis_client.put_record(
                StreamName=kinesis_name, Data=to_bytes(json.dumps(message)), PartitionKey="custom"
            )
            def verify_failure_received():