### 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`
)
    elif integration_type in ["HTTP_PROXY", "HTTP"]:
        if ":servicediscovery:" in uri:
            
            client = aws_stack.connect_to_service("servicediscovery")
            service_id = uri.split("/")[-1]
            instances = client.list_instances(ServiceId=service_id)["Instances"]
            instance = (instances or [None])[0]
            if instance and instance.get("Id"):
                uri = "http://%s/%s" % (instance["Id"], invocation_path.lstrip("/"))
        
        invocation_context.context = helpers.get_event_request_context(invocation_context)
        invocation_context.stage_variables = helpers.get_stage_variables(invocation_context)
        request_templates = RequestTemplates()
        payload = request_templates.render(invocation_context)
        if isinstance(payload, dict):
            payload = json.dumps(payload)
        uri = apply_request_parameters(
            uri,
            integration=integration,
            path_params=path_params,
            query_params=query_string_params,
        )
        result = requests.request(method=method, url=uri, data=payload, headers=headers)
        
        invocation_context.response = result
        response_templates = ResponseTemplates()
        response_templates.render(invocation_context)
        return invocation_context.response
    elif integration_type == "MOCK":