### 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_
helpers.set_api_id_stage_invocation_path(invocation_context)
    try:
        return invoke_rest_api(invocation_context)
    except AuthorizationError as e:
        api_id = invocation_context.api_id
        return make_error_response("Not authorized to invoke REST API %s: %s" % (api_id, e), 403)
def invoke_rest_api(invocation_context: ApiInvocationContext):
    invocation_path = invocation_context.path_with_query_string
    raw_path = invocation_context.path or invocation_path
    method = invocation_context.method
    headers = invocation_context.headers
    
    authorize_invocation(invocation_context)
    extracted_path, resource = helpers.get_target_resource_details(invocation_context)
    if not resource:
        return make_error_response("Unable to find path %s" % invocation_context.path, 404)
    
    validator = RequestValidator(invocation_context)
    if not validator.is_request_valid():
        return make_error_response("Invalid request body", 400)
    api_key_required = resource.get("resourceMethods", {}).get(method, {}).get("apiKeyRequired")
    if api_key_required and not is_api_key_valid(invocation_context):
        return make_error_response("Access denied - invalid API key", 403)
    resource_methods = resource.get("resourceMethods", {})
    resource_method = resource_methods.get(method, {})
    if not resource_method:
        
        
        resource_method = resource_methods.get("ANY", {}) or resource_methods.get(
            "X-AMAZON-APIGATEWAY-ANY-METHOD", {}