While I know it would be time containing to make all three house leaders their own thing: It would be worth it just for the ****storm it would give the Smash community lmao.
Heck in a previous post, I even thought up perfect color alts for them referencing their housemates (which coincidentally enough equal to 8 counting themselves). So that means:
Edelgard could get: Default, Hubert, Bernadetta, Dorothea, Petra, Ferdinand, Caspar, and Linhardt
Dmitri could get: Default, Dedue, Ashe, Felix, Sylvain, Ingrid, Annette, and Mercedes
And Claude could get: Default, Hilda, Lorenz, Ignatz, Raphael, Leonie, Lysithea, and Marianne
Pretty much, her significance to the plot kind of means it's actually fair representation to the routes, using her as a rep. It'd only really be Crimson Flower bias if she had Byleth's engagement ring and/or her positive traits got played up more.
What are the chances of Byleth getting "chance another dayed" as part of Edie's final smash?
Considering how Smash condenses (or in some cases flanderizes) the characterizations of the fighters to their base core, if Edelgard does get in: I see her simply being depicted as a standard stern and driven individual (aka her default characterization) with no additional characterizations unique to a particular route.