### 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_
)
    method_integration = resource_method.get("methodIntegration")
    if not method_integration:
        if method == "OPTIONS" and "Origin" in headers:
            
            return get_cors_response(headers)
        return make_error_response(
            "Unable to find integration for: %s %s (%s)" % (method, invocation_path, raw_path),
            404,
        )
    
    invocation_context.resource_path = extracted_path
    invocation_context.integration = method_integration
    return invoke_rest_api_integration(invocation_context)
def invoke_rest_api_integration(invocation_context: ApiInvocationContext):
    try:
        response = invoke_rest_api_integration_backend(invocation_context)
        
        
        invocation_context.response = response
        response = apply_response_parameters(invocation_context)
        return response
    except ApiGatewayIntegrationError as e:
        LOG.warning(
            "Error while invoking integration for ApiGateway ID %s: %s",
            invocation_context.api_id,
            e,
            exc_info=LOG.isEnabledFor(logging.DEBUG),
        )
        return e.to_response()