### 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_
except Exception as e:
        msg = f"Error invoking integration for API Gateway ID '{invocation_context.api_id}': {e}"
        LOG.exception(msg)
        return make_error_response(msg, 400)
def invoke_rest_api_integration_backend(invocation_context: ApiInvocationContext):
    
    invocation_path = invocation_context.path_with_query_string
    method = invocation_context.method
    headers = invocation_context.headers
    resource_path = invocation_context.resource_path
    integration = invocation_context.integration
    
    relative_path, query_string_params = extract_query_string_params(path=invocation_path)
    integration_type_orig = integration.get("type") or integration.get("integrationType") or ""
    integration_type = integration_type_orig.upper()
    integration_method = integration.get("httpMethod")
    uri = integration.get("uri") or integration.get("integrationUri") or ""
    try:
        invocation_context.path_params = extract_path_params(
            path=relative_path, extracted_path=resource_path
        )
    except Exception:
        invocation_context.path_params = {}
    if (uri.startswith("arn:aws:apigateway:") and ":lambda:path" in uri) or uri.startswith(
        "arn:aws:lambda"
    ):
        if integration_type == "AWS_PROXY":
            return LambdaProxyIntegration().invoke(invocation_context)
        elif integration_type == "AWS":
            return LambdaIntegration().invoke(invocation_context)