Describe the bug

I was following this documentation section, where there's a workaround for multiple stack describing api gateway resources:

Breaking up Methods and Resources across Stacks
It is fairly common for REST APIs with a large number of Resources and Methods to hit the CloudFormation limit of 500 resources per stack.

So I broke my stack in several nested stacks and imported back my api in each nested stack using the ``fromRestApiAttributes``` method. The correct object was found but not all the configuration appeared.

Expected Behavior

I was expecting the full object as set during the API creation in the main stack.

Current Behavior

Dumping the object, I had the following:

{
...
    defaultIntegration: undefined,
    defaultMethodOptions: undefined,
    defaultCorsPreflightOptions: undefined
...
}

instead of:

{
...
    defaultIntegration: undefined,
    defaultMethodOptions: undefined,
    defaultCorsPreflightOptions: { allowOrigins: [Array], allowMethods: [Array] },
...
}

Reproduction Steps

The following piece of code as mentionned in the documentation:
https://docs.aws.amazon.com/cdk/api/v2/docs/aws-cdk-lib.aws_apigateway-readme.html#breaking-up-methods-and-resources-across-stacks

Possible Solution

No response

Additional Information/Context

No response

CDK CLI Version

2.127.0 (build 6c90efc)

Framework Version

No response

Node.js Version

v21.1.0

OS

Mac

Language

TypeScript

Language Version

No response

Other information

No response