### 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_
elif integration_type == "AWS":
        if "kinesis:action/" in uri:
            return KinesisIntegration().invoke(invocation_context)
        if "states:action/" in uri:
            return StepFunctionIntegration().invoke(invocation_context)
        if ":dynamodb:action" in uri:
            return DynamoDBIntegration().invoke(invocation_context)
        if "s3:path/" in uri or "s3:action/" in uri:
            return S3Integration().invoke(invocation_context)
        if integration_method == "POST" and ":sqs:path" in uri:
            return SQSIntegration().invoke(invocation_context)
        if method == "POST" and ":sns:path" in uri:
            return SNSIntegration().invoke(invocation_context)
        if (
            method == "POST"
            and uri.startswith("arn:aws:apigateway:")
            and "events:action/PutEvents" in uri
        ):
            return EventBridgeIntegration().invoke(invocation_context)
    elif integration_type in ["HTTP_PROXY", "HTTP"]:
        return HTTPIntegration().invoke(invocation_context)
    elif integration_type == "MOCK":
        return MockIntegration().invoke(invocation_context)
    if method == "OPTIONS":
        
        return get_cors_response(headers)
    raise Exception(
        f'API Gateway integration type "{integration_type}", method "{method}", URI "{uri}" not yet implemented'
    )