Tathanen
Get Inside Her!
Again the only thing we know is that Itachi knew that Tobi wasn't a normal ninja and the fact that he thought he was Madara, he never worked with him closely.
Itachi worked directly with Tobi to exterminate the Uchiha clan, after seeking him out personally. I think that was KINDA close.
You seem to forget that Kamui isn't warping the person who uses it out only but also selective objects like Kakashi used, he could've easily warp out only the flames and let them burn in the other dimension instead of essentially trashing an eye.
I don't really buy that one. He uses Kamui in two ways. One, the passive one where he toggles parts of his body to another dimension, and two, the active one where he chooses a point in space and warps away its contents. The flames were burning all over his shoulder. He'd have to use Kamui in the second way to target that, and he can't just "target flames" and remove them, they're spread over the fabric of his outfit there. Even if we assume that he COULD target all of the flame, and only the flame, and teleport it away, there's a period of "ugh agh oh fuck" coming from him in the darkness, then silence, then he walks back perfectly fine. No burnt-off clothes, no damage whatsoever. He acts like he totally defeated the attack, not like he just removed it. I do think that his "fortunately I managed to keep a few secrets even from him" is a weird thing to say if he used Izanagi, since Izanagi should be something Itachi would expect, but he still doesn't really look or act like a man just set on fire. He also suggests he'd be dead if not for the secret thing he was able to do, and even if Kamui hadn't been an option, he could've always Izanagi'd. Though that presupposes that Kishimoto had even thought of Izanagi at that time. I gotta believe he had, Danzo had been around for long enough, and that character design was probably built with foreknowledge of the sharingan arm.
..it coulda been Kamui. But I think it's kinda inconsistent either way.