Just had a thought
Wouldn't it have been funny if Byleth had recognized that Rhea was Sothis's daughter during the scene in the holy tomb and Byleth just pretended to be Sothis in order to punish her disobedient child? Something like "Of course I'd side with Edelgard. You're the one who tried to bring me back when I never asked nor really wanted you to!* Problem solved, and Edelgard/Rhea and Byleth go to destroy the Agarthans once and for all. Dimitri would need another minute to think about it, but all Edelgard had to say was that she knew who was actually responsible for the Tragedy of Duscur.
It's clear from the BL version of WC that Dimitri harbored his own suspicions about Arundel/Thales.
And Claude could have easily been persuaded, had
!BylethSothis simply gotten on Rhea's case for closing Fodlan out and restricting information.
Golden Ending. Boom.