### 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`
func_arn,
                relative_path,
                payload,
                stage,
                api_id,
                headers,
                is_base64_encoded=invocation_context.is_data_base64_encoded,
                path_params=path_params,
                query_string_params=query_string_params,
                method=method,
                resource_path=resource_path,
                request_context=invocation_context.context,
                stage_variables=invocation_context.stage_variables,
            )
            if isinstance(result, FlaskResponse):
                response = flask_to_requests_response(result)
            elif isinstance(result, Response):
                response = result
            else:
                response = LambdaResponse()
                response.headers.update({"content-type": "application/json"})
                parsed_result = (
                    result if isinstance(result, dict) else json.loads(str(result or "{}"))
                )
                parsed_result = common.json_safe(parsed_result)
                parsed_result = {} if parsed_result is None else parsed_result
                response.status_code = 200
                response._content = parsed_result
                update_content_length(response)