### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

When adding a custom id to an apigateway V1 via tags in aws-cdk the id will be applied and deploying returns the url including the custom id (https://<custom-id>.execute-api.localhost.localstack.cloud:4566/prod/). Querying this url however will return a 404 response. 

The url generated when no custom id is provided works with issue.

### Expected Behavior

The custom id is applied and querying works with the url including the custom id.

### 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 -d

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)
*AWS CDK (python)*
```python
# test-stack.py
from aws_cdk import (
    Stack,
    aws_lambda,
    aws_apigateway as api_gateway,
    Tags,
)
from pathlib import Path
from constructs import Construct


class TestStack(Stack):
    def __init__(self, scope: Construct, construct_id: str, **kwargs) -> None:
        super().__init__(scope, construct_id, **kwargs)

        my_lambda = aws_lambda.Function(
            self,
            "test-lambda",
            code=aws_lambda.Code.from_asset("lambdas"),
            runtime=aws_lambda.Runtime.PYTHON_3_9,
            handler="test_lambda.handle",
        )

        api = api_gateway.RestApi(
            self,
            "Api",
            rest_api_name="apiName",
        )
        Tags.of(api).add(
            "_custom_id_", "myCustomId"
        )  # makes api id and therefore url static

        api.root.add_method(
            "GET", api_gateway.LambdaIntegration(my_lambda)  # type: ignore
        )

```
```python
# lambdas/test_lambda.py
def handle(event, context):
    return {"statusCode": 200, "body": {}}
```

Deploy outputs: https://myCustomId.execute-api.localhost.localstack.cloud:4566/prod/
Querying this url returns 404

![localstack-bug-postman](https://user-images.githubusercontent.com/48515433/201137013-0e0f1161-7818-427a-88d8-a12e9c2b2199.JPG)

### Environment

```markdown
- OS: Windows 10
- LocalStack: latest (10.11.2022)
- aws-cdk@2.44.0
```


### Anything else?

I found [#4499](https://github.com/localstack/localstack/issues/4499), but from the deploy output and awslocal it seems that the Api Gateway should have been created with the custom id but cannot be queried.  

---
Result without custom id:

![localstack-bug-postman-expected](https://user-images.githubusercontent.com/48515433/201140708-84f49aa2-cd3f-49e1-8947-9b6572941630.JPG)

return method == "POST" and bool(re.match(PATH_REGEX_TEST_INVOKE_API, path))
def get_stage_variables(context: ApiInvocationContext) -> Optional[Dict[str, str]]:
    if is_test_invoke_method(context.method, context.path):
        return None
    if not context.stage:
        return {}
    _, region_name = get_api_account_id_and_region(context.api_id)
    api_gateway_client = aws_stack.connect_to_service("apigateway", region_name=region_name)
    try:
        response = api_gateway_client.get_stage(restApiId=context.api_id, stageName=context.stage)
        return response.get("variables")
    except Exception:
        LOG.info("Failed to get stage %s for API id %s", context.stage, context.api_id)
        return {}
def gateway_response_to_response_json(item, api_id):
    base_path = "/restapis/%s/gatewayresponses" % api_id
    item["_links"] = {
        "self": {"href": "%s/%s" % (base_path, item["responseType"])},
        "gatewayresponse:put": {
            "href": "%s/{response_type}" % base_path,
            "templated": True,
        },
        "gatewayresponse:update": {"href": "%s/%s" % (base_path, item["responseType"])},
    }
    item["responseParameters"] = item.get("responseParameters", {})
    item["responseTemplates"] = item.get("responseTemplates", {})
    return item
def get_gateway_responses(api_id):