### 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`
headers = aws_stack.mock_aws_request_headers(
                service="kinesis", region_name=invocation_context.region_name
            )
            headers["X-Amz-Target"] = target
            result = common.make_http_request(
                url=config.service_url("kineses"), data=payload, headers=headers, method="POST"
            )
            
            invocation_context.response = result
            response_templates = ResponseTemplates()
            response_templates.render(invocation_context)
            return invocation_context.response
        elif "states:action/" in uri:
            action = uri.split("/")[-1]
            if APPLICATION_JSON in integration.get("requestTemplates", {}):
                request_templates = RequestTemplates()
                payload = request_templates.render(invocation_context)
                payload = json.loads(payload)
            else:
                
                payload = json.loads(data.decode("utf-8"))
            client = aws_stack.connect_to_service("stepfunctions")
            if isinstance(payload.get("input"), dict):
                payload["input"] = json.dumps(payload["input"])
            
            method_name = (
                camel_to_snake_case(action) if action != "StartSyncExecution" else "start_execution"
            )
            try:
                method = getattr(client, method_name)