Handling Split Domains: Any Tips?

I'm currently the sole admin for a small non-profit that has teamed up with a larger organization. We're transitioning to a new local domain that integrates with Entra, aiming to utilize the security features necessary for compliance with the larger org's requirements. My users log into ad.myorg.com, but we access Office 365 via largeorg.com, for which I have no admin control. Usually, everything works fine, but occasionally users face issues where Office 365 stubbornly attempts to log in with their ad.myorg.com credentials instead of their largerorg.com ones, especially with this new domain setup. The larger org has offered to include us as a tenant in their Active Directory, but our Director prefers to keep things separate. I'm wondering if anyone else has dealt with a scenario like this and if there's a more efficient way to manage it? Thanks in advance for any insights!