Hi Julian! I hope you are having a good day.
An IDP file will need to be configured for each subsequent subdomain that needs to use the inherited SAML domain setting. Also, an app will need to be created for each domain since the app will have a unique ID for each domain.
Very curious as to why this application behaves like this? That is a lot for a district to maintain. Why not enable the SSO to pass through to the subdomains?