### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

APiGateway Model that being created by CDK project stores the JSON schema with a single quotes instead of double quotes which leading to an error in parsing this json object. 

you can refere to this article to see the steps 

https://dev.to/dvddpl/reduce-lambda-invocations-and-code-boilerplate-by-adding-a-json-schema-validator-to-your-api-gateway-15af

and it works fine on AWS account 

also you can look on the attached image how localstack stores the JSON object with a single quotes, I tried to edit it manually through UI but it return it back with single quotes

![Screen Shot 2022-10-18 at 4 28 08 PM](https://user-images.githubusercontent.com/106184681/196769273-0834865a-2fb3-4efc-b3a9-9da6b2ad511d.png)

### Expected Behavior

store the json schema in correct JSON format

### How are you starting LocalStack?

With the `localstack` script

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

    localstack start

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

    cdklocal deploy


### Environment

```markdown
- OS:OSX
- LocalStack: 1.0.4
```


### Anything else?

_No response_
httpMethod=method,
                    **kwargs,
                )
                default_params = (
                    "responseParameters",
                    "responseTemplates",
                )
                for integration_response in integration_responses:
                    integration_response["statusCode"] = str(integration_response["statusCode"])
                    for param in default_params:
                        integration_response[param] = integration_response.get(param) or {}
                    apigateway.put_integration_response(
                        restApiId=api_id,
                        resourceId=res_id,
                        httpMethod=method,
                        **keys_to_lower(integration_response),
                    )
            responses = props.get("MethodResponses") or []
            for response in responses:
                api_id = props["RestApiId"]
                res_id = props["ResourceId"]
                apigateway.put_method_response(
                    restApiId=api_id,
                    resourceId=res_id,
                    httpMethod=props["HttpMethod"],
                    statusCode=str(response["statusCode"]),
                    responseParameters=response.get("responseParameters") or {},
                    responseModels=response.get("responseModels") or {},
                )
        def _handle_result(result, resource_id, resources, resource_type):