### 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_
rest_api: RestApi = call_moto(context)
        remove_empty_attributes_from_rest_api(rest_api)
        return rest_api
    def update_rest_api(
        self,
        context: RequestContext,
        rest_api_id: String,
        patch_operations: ListOfPatchOperation = None,
    ) -> RestApi:
        rest_api = get_moto_rest_api(context, rest_api_id)
        fixed_patch_ops = []
        binary_media_types_path = "/binaryMediaTypes"
        
        for patch_op in patch_operations:
            patch_op_path = patch_op.get("path", "")
            
            
            
            if patch_op_path.startswith(binary_media_types_path):
                if patch_op_path == binary_media_types_path:
                    raise BadRequestException(f"Invalid patch path {patch_op_path}")
                value = patch_op_path.rsplit("/", maxsplit=1)[-1]
                path_value = value.replace("~1", "/")
                patch_op["path"] = binary_media_types_path
                if patch_op["op"] == "add":
                    patch_op["value"] = path_value
                elif patch_op["op"] == "remove":
                    remove_index = rest_api.binaryMediaTypes.index(path_value)