### 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`
mock_integration = MockIntegration()
        return mock_integration.invoke(invocation_context)
    if method == "OPTIONS":
        
        return get_cors_response(headers)
    raise Exception(
        'API Gateway integration type "%s", method "%s", URI "%s" not yet implemented'
        % (integration_type, method, uri)
    )
def apply_request_response_templates(
    data: Union[Response, bytes],
    templates: Dict[str, str],
    content_type: str = None,
    as_json: bool = False,
):
    """Apply the matching request/response template (if it exists) to the payload data and return the result"""
    content_type = content_type or APPLICATION_JSON
    is_response = isinstance(data, Response)
    templates = templates or {}
    template = templates.get(content_type)
    if not template:
        return data
    content = (data.content if is_response else data) or ""
    result = VtlTemplate().render_vtl(template, content, as_json=as_json)
    if is_response:
        data._content = result
        update_content_length(data)
        return data
    return result