### 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`
target_uri,
            path=lambda_resource,
            stage_name=TEST_STAGE_NAME,
        )
        api_id = result["id"]
        url = path_based_url(api_id=api_id, stage_name=TEST_STAGE_NAME, path=lambda_path)
        result = safe_requests.get(url)
        assert result.status_code == 502
class TestKinesisSource:
    def test_create_kinesis_event_source_mapping(
        self,
        create_lambda_function,
        lambda_client,
        kinesis_client,
        kinesis_create_stream,
        lambda_su_role,
        wait_for_stream_ready,
        logs_client,
    ):
        function_name = f"lambda_func-{short_uid()}"
        stream_name = f"test-foobar-{short_uid()}"
        record_data = "hello"
        num_events_kinesis = 10
        try:
            create_lambda_function(
                func_name=function_name,
                handler_file=TEST_LAMBDA_PYTHON_ECHO,
                runtime=LAMBDA_RUNTIME_PYTHON36,
                role=lambda_su_role,
            )