Given the story that's been told, I don't think Kenny/Cody really needs the US Title on the line. Cody's been questioning Kenny's leadership for pretty much the entire time he's been in Bullet Club and he challenged Ibushi for Wrestle Kingdom seemingly just to get under Kenny's skin.
Maybe their eventual match will be tied with a loser leaves the Bullet Club stip. A lot of people are taking this angle as Kenny's already been booted out but if that were the case I think they all would've partook in beating down Kenny. I actually thought that's what was coming when Kenny shoved down Matt Jackson.
As it was though only Hangman Page seemed like he was actively siding with Cody, helping to hold him up for the chair attack. The Bucks seemed annoyed at Kenny in the moment but mostly just wanted Kenny and Cody to sort their shit, and Marty Scurll was actively trying to play peacemaker the whole time.
I'd have to guess Marty and the Bucks will formally turn on Kenny before long. Losing Marty and the Bucks would pretty much kill Bullet Club and seeing how much Bullet Club merch those guys are still selling, I can't see that happening.
Kenny will get a new line of merch for Hot Topic and it'll work out. He'll have a few big tag matches with Ibushi, and for multi-man tags he'll probably get teamed with the Taguchi Japan guys since that's pretty much just an open door "stable" for people not affiliated with the four main groups.
As for Jay White, I'm guessing Hangman will be his first challenger. Hangman's not really done anything other than tag matches in New Japan as best I can tell, but if he's gonna be in "The Elite" portion of the stable it'd be best to get him some exposure in singles matches.