### 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`
% (uri, method)
        )
    elif integration_type == "AWS_PROXY":
        if uri.startswith("arn:aws:apigateway:") and ":dynamodb:action" in uri:
            
            table_name = uri.split(":dynamodb:action")[1].split("&Table=")[1]
            action = uri.split(":dynamodb:action")[1].split("&Table=")[0]
            if "PutItem" in action and method == "PUT":
                response_template = response_templates.get("application/json")
                if response_template is None:
                    msg = "Invalid response template defined in integration response."
                    LOG.info("%s Existing: %s", msg, response_templates)
                    return make_error_response(msg, 404)
                response_template = json.loads(response_template)
                if response_template["TableName"] != table_name:
                    msg = "Invalid table name specified in integration response template."
                    return make_error_response(msg, 404)
                dynamo_client = aws_stack.connect_to_resource("dynamodb")
                table = dynamo_client.Table(table_name)
                event_data = {}
                data_dict = json.loads(data)
                for key, _ in response_template["Item"].items():
                    event_data[key] = data_dict[key]
                table.put_item(Item=event_data)
                response = requests_response(event_data)
                return response
        else:
            raise Exception(
                'API Gateway action uri "%s", integration type %s not yet implemented'
                % (uri, integration_type)