### 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_
context: RequestContext, action: str, parameters: ServiceRequest
) -> RequestContext:
    ctx = create_aws_request_context(
        service_name=context.service.service_name,
        action=action,
        parameters=parameters,
        region=context.region,
    )
    ctx.request.headers.update(context.request.headers)
    ctx.account_id = context.account_id
    return ctx
def _patch_api_gateway_entity(entity: Any, patch_operations: ListOfPatchOperation):
    if not isinstance(entity.__dict__, DelSafeDict):
        entity.__dict__ = DelSafeDict(entity.__dict__)
    not_supported_attributes = {"/id", "/region_name", "/create_date"}
    model_attributes = list(entity.__dict__.keys())
    for operation in patch_operations:
        path_start = operation["path"].strip("/").split("/")[0]
        path_start_usc = camelcase_to_underscores(path_start)
        if path_start not in model_attributes and path_start_usc in model_attributes:
            operation["path"] = operation["path"].replace(path_start, path_start_usc)
        if operation["path"] in not_supported_attributes:
            raise BadRequestException(f"Invalid patch path {operation['path']}")
    apply_json_patch_safe(entity.__dict__, patch_operations, in_place=True)
def to_authorizer_response_json(api_id, data):