...If Kimblee-san is the real False Seer, that means whomever wrote that note isn't. They couldn't claim that Kimblee-san is a demon in that case, because there's a chance that they're wrong. So the only way that note makes any sense is if the person writing it couldn't be sure.

[...]

...Or perhaps if they can be sure, and know for a fact that Kimblee-san isn't a demon at all. "Outing" Kimblee-san in that case would destroy the ruse and their word couldn't be trusted again.

The demons would know. In that case, this would be a solid argument for why Kimblee-san isn't one of theirs, and why the dove might be.