Inoichi might be able to tell that something is a little hinky with the story, might even know that the story is A Complete Lie, but he trusts Shikaku and respects his right to have clan secrets. The girl is Definitely A Nara and Shikaku is sure she can be trusted so like... that's good enough for Inoichi.

Oh man, this trust so so good. Especially when, you know, it turns out Shikako is actually pretty strong and Inoichi just has to keep looking at the situation like... 'i don't know what's going on here but I'm sure it's fine'.

Also, plus, he's definitely seen Shikako select her acquaintances very very carefully (mentioned up thread: Aoba -> Anko -> Kurenai -> probably Gai, Genma, Ebisu, etc.) so if she's hanging with this Uchiha

Oh man, related to the 'Shisui only had one (1) friend' comment further down... Shisui making friends with the Sp Jounin crew? Plz. Aoba would be totally willing to get Uchiha gossip first make new friends.