### 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_
if len(path.split("/")) > 3 or not any(
                path.startswith(s_path) for s_path in UPDATE_METHOD_PATCH_PATHS["supported_paths"]
            ):
                raise BadRequestException(f"Invalid patch path {path}")
            
            op_supported_path = UPDATE_METHOD_PATCH_PATHS.get(op, [])
            if not any(path.startswith(s_path) for s_path in op_supported_path):
                continue
            value = patch_operation.get("value")
            if op not in ("add", "replace"):
                
                applicable_patch_operations.append(patch_operation)
                continue
            if path == "/authorizationType" and value in ("CUSTOM", "COGNITO_USER_POOLS"):
                modifying_auth_type = True
            elif path == "/authorizerId":
                modified_authorizer_id = value
            if any(
                path.startswith(s_path) for s_path in ("/apiKeyRequired", "/requestParameters/")
            ):
                patch_op = {"op": op, "path": path, "value": str_to_bool(value)}
                applicable_patch_operations.append(patch_op)
                continue
            elif path == "/requestValidatorId" and value not in rest_api.validators:
                if not value:
                    
                    patch_op = {"op": "remove", "path": path, "value": value}
                    applicable_patch_operations.append(patch_op)
                    continue