### 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`
"""
    Handles request template rendering
    """
    def render(self, api_context: ApiInvocationContext) -> Union[bytes, str]:
        LOG.info(
            "Method request body before transformations: %s", to_str(api_context.data_as_string())
        )
        request_templates = api_context.integration.get("requestTemplates", {})
        template = request_templates.get(APPLICATION_JSON, {})
        if not template:
            return api_context.data_as_string()
        variables = self.build_variables_mapping(api_context)
        result = self.render_vtl(template, variables=variables)
        LOG.info(f"Endpoint request body after transformations:\n{result}")
        return result
class ResponseTemplates(Templates):
    """
    Handles response template rendering
    """
    def render(self, api_context: ApiInvocationContext, **kwargs) -> Union[bytes, str]: