### 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_
@staticmethod
    def cloudformation_type():
        return "AWS::ApiGateway::BasePathMapping"
    def fetch_state(self, stack_name, resources):
        resources = (
            connect_to()
            .apigateway.get_base_path_mappings(domainName=self.props.get("DomainName"))
            .get("items", [])
        )
        comparable = (
            [self.props.get("BasePath")] if self.props.get("BasePath") else [None, "", "(none)"]
        )
        return next(iter(res for res in resources if res.get("basePath") in comparable))
    @classmethod
    def get_deploy_templates(cls):
        def _create_base_path_mapping(logical_resource_id: str, resource: dict, stack_name: str):
            provider = cls(resource)
            props = provider.props
            kwargs = {
                "domainName": props.get("DomainName"),
                "restApiId": props.get("RestApiId"),
                **({"basePath": props.get("BasePath")} if props.get("BasePath") else {}),
                **({"stage": props.get("Stage")} if props.get("Stage") else {}),
            }
            return connect_to().apigateway.create_base_path_mapping(**kwargs)
        def _handle_result(result, resource_id, resources, resource_type):