### 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`
actual_record_ids.append(actual_record_id)
                actual_record_ids.sort()
                assert actual_record_ids == [i for i in range(num_records_per_batch)]
        finally:
            lambda_client.delete_event_source_mapping(UUID=uuid)
    def test_disable_kinesis_event_source_mapping(
        self,
        lambda_client,
        kinesis_client,
        create_lambda_function,
        kinesis_create_stream,
        wait_for_stream_ready,
        logs_client,
        lambda_su_role,
    ):
        function_name = f"lambda_func-{short_uid()}"
        stream_name = f"test-foobar-{short_uid()}"
        num_records_per_batch = 10
        try:
            create_lambda_function(
                handler_file=TEST_LAMBDA_PYTHON_ECHO,
                func_name=function_name,
                runtime=LAMBDA_RUNTIME_PYTHON36,
                role=lambda_su_role,
            )
            kinesis_create_stream(StreamName=stream_name, ShardCount=1)
            stream_arn = kinesis_client.describe_stream(StreamName=stream_name)[
                "StreamDescription"
            ]["StreamARN"]
            wait_for_stream_ready(stream_name=stream_name)