#the Kingdom isn't a potential threat to Claude; it has done nothing to garner any attention (positive or negative) from the Alliance

[butwhatifidothis]

Do you think it'd be in character for Claude to fake an alliance with Edgey in GW Pt 2 so that he could undermine her while seeking an alliance with the Kingdom behind her back? I mean, that way, he could ask Rhea directly about what Edgey told him instead of blindly believing her.

It would be more in character, but not as much as it looks at first.

Because yes, Claude is very familiar with not exactly telling the truth, and he is also familiar with working around people who inherently distrust him. And something somewhat like this kinda happens in 3H, with Erwin - Claude has Nader create a distraction for the army to pass through Gloucester territory to Myrddin so that Erwin's forces don't intercept them (he does this in all the routes in fact, save for of course CF). So it's not like he isn't capable of some form of trickery, even when it comes to people he considers allies.

What becomes an issue though is that those sort of trickery means are mostly used for last-resort, fairly specific situations - Plan A wasn't to make Erwin think his lands are being attacked, it was the only option Claude had available due to the very landscape of Leicester necessitating going through Gloucester territory, and due to Erwin's stubbornness about aligning with the Empire (whether or not those reasons were self-serving or for self-preservation). Again, with Myrddin we see that he's willing to openly help the Kingdom/Church, despite the act of doing so breaking any neutrality the Alliance could have held onto. Communication with Erwin had already failed (due to, again, his stubbornness to align with the Empire), but Claude also explicitly says that he doesn't want to engage in actual combat with House Gloucester, so lies were the only way to go forward.

With the Empire, however, there was never any worry about engaging in combat with them - they invaded his lands, and he fought back. If Edelgard had started with negotiations with Claude first, that'd be one thing - if there was an attempt at an assurance for peace from Edelgard's end to start everything off, it'd make the idea of him joining her to betray her later have some ground to stand on logically. But with him doing absolutely nothing to provoke her she attacks his lands, which has two reasons for her doing so in specifically Claude's mind: taking Garreg Mach to use as a base, and getting rid of the Alliance and Kingdom at the same time.

To the first reason: while Garreg Mach is a good base, it's arguable that having to defeat the Alliance on top of fighting the Kingdom for it makes taking the base moot to begin with. Even for the symbolic nature of it, since Rhea has already been booted out at this point. The trouble would be more than its worth, essentially. And the second reason is... fairly obvious as to why that isn't gonna work out lmao.

Then there's also also the fact that Claude was handling the Empire pretty damn well - he is a young leader and is tricksty in his antics which means skepticism from his more experienced peers, the Alliance is known for its weaker military strength, and he had to deal with Shahid looming over his head, yet Claude still manages to hold off the Empire pretty effectively. So with Edelgard invading Claude's lands off rip and his shown proficiency at fighting back against her without much non-Alliance help, there's no benefit to siding with her, even falsely - none that would outweigh the benefits of siding with a party that has given him no misgivings, in any case. She's not a force that is too big to take on from the outside and so needs to be taken out from the inside, so going through the effort of getting inside is meaningless.

So, would falsely siding with Edelgard while secretly siding with the Kingdom be more in character? It certainly can be, if one plays the cards right. But given how Claude's handling of the Empire and Edelgard's actions against the Alliance are written, it'd be far more in character for him to just side with the Kingdom outright at that point