Since there is going to be direct dealings with the Shogun, I'm not sure Fatui would be all that welcome considering their recent history in Inazuma. Really for someone like Childe, it depends on what characters others would want to play. For example, if someone played Aether/Lumine, they'd be much more willing to vouch for Childe's willingness to help, even if he's just there for a good fight. But without The Traveler, people aren't going to trust him, and even less so some Fatui The Traveler doesn't even know. Now if your character was more, secretly a Fatui agent... something like that could possibly work, especially if they are from Inazuma. 

So I'd say who you play may depend on who others decide to play. Someone like Itto is more neutral, while someone like Ayato already has a chosen 'side' so to speak.