### 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_
raise BadRequestException("Invalid Request Validator identifier specified")
            elif path.startswith("/requestModels/"):
                if value != EMPTY_MODEL and value not in rest_api.models:
                    raise BadRequestException(f"Invalid model identifier specified: {value}")
            applicable_patch_operations.append(patch_operation)
        if modifying_auth_type:
            if not modified_authorizer_id or modified_authorizer_id not in rest_api.authorizers:
                raise BadRequestException(
                    "Invalid authorizer ID specified. "
                    "Setting the authorization type to CUSTOM or COGNITO_USER_POOLS requires a valid authorizer."
                )
        elif modified_authorizer_id:
            if moto_method.authorization_type not in ("CUSTOM", "COGNITO_USER_POOLS"):
                
                
                applicable_patch_operations = [
                    op for op in applicable_patch_operations if op.get("path") != "/authorizerId"
                ]
        
        _patch_api_gateway_entity(moto_method, applicable_patch_operations)
        
        if had_req_params and len(moto_method.request_parameters) == 0:
            moto_method.request_parameters = None
        if had_req_models and len(moto_method.request_models) == 0:
            moto_method.request_models = None
        response = moto_method.to_json()
        remove_empty_attributes_from_method(response)
        remove_empty_attributes_from_integration(response.get("methodIntegration"))
        return response
    def delete_method(