the attack in question is near 2:40 here https://youtu.be/-J7V4YykpY4?t=160 (episode 456?)
how did he survive that attack? Shouldn't Amaterasu have burned his body down to ash?
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While the scene you are asking about is partially filler, we do know some elements about both Amaterasu and the beast's cloak that can provide an answer.
The tailed beast's cloak, while active, provides raid healing effects. So while Yagura is in the beast form, the beast's chakra was constantly healing while Amaterasu was burning. Eventually Amaterasu's power overtook Yagura's beast forum, thus forcing him to revert back his human state.
When Amaterasu is active, there are only a handful of methods to extinguish the flames. It will eventually burn out after 7 days and nights or if the user recalls the flames; whichever comes first. So after Yagura reverted back to his human state, Itachi put out the flames to not actually kill him, but to immobilize him.