### 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`
}
            invocation_context.context["authorizer"] = authorizer_context
        payload = self.request_templates.render(invocation_context)
        
        result = lambda_api.process_apigateway_invocation(
            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=invocation_context.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()
            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
            keys = parsed_result.keys()
            if not ("statusCode" in keys and "body" in keys):