SharkLord
Smash Hero
I've been debating Byleth vs Edelgard with my brother a bit lately. Putting aside the rather unlikely scenario that they go out of their way to replace a veteran, the conclusion we reached was "Do we upset Smash fans or Fire Emblem fans?"Old grudges die hard. So not really.
I have to say, I don't see them cutting Byleth to add another 3H character(and when people say 'another 3H character', they inevitably mean Edelgard.) Like yeah, Edelgard is more popular(and controversial) within the FE fanbase. But Byleth is still a very popular character, the game's main protagonist, a neutral character who doesn't favour one route over another, and is already in smash, so they don't have to make a new moveset.
Like you mentioned, Byleth isthe player character of Three Houses and therefore the safest, most neutral option from Three Houses, which puts a bit emphasis on multiple routes. (And like you mentioned they're already in the game, which most likely gives them priority by default). That said, for a sizeable contingent of Smash fans, Byleth is Yet Another Blue Haired Fire Emblem Swordfighter, and even if you can spin something unique out of them, they still carry that stigma. To avoid that, Sakurai gave them the other Relics, but that means each weapon doesn't get much room to breathe and completely ignores the Divine Pulse, upsetting Fire Emblem fans as well.
Edelgard would be more distinct in her design and moveset. She covers a new weapon type, represents Armor units, could be counted as a villain rep depending on the route (though that's a big can of worms apparently), is an active player in all routes of Three Houses, and I believe she's the most popular character in Three Houses (Granted, that statistic is from Heroes' Choose Your Legends vote, but that was during a big surge of Three Houses newcomers so I think it's a bit more accurate?). However, that means favoring one route over all the others, and the Three Houses Lords are usually shown together to avoid that. In addition, I believe Edelgard is a rather divisive character among the Three Houses fandom, so if you add her over anyone else... Well, there's a reason we say "four more years of Three Houses discourse."
Clearly, the answer is to add Lysithea instead and let her pop Shields from halfway across the stage with Dark Spikes T