Of course she won't look in to it.
1. She doesn't know the MC. No reason to trust him - considering how sheltered and asocial Sarah is, there is a 50/50 chance her judgement of him is clouded.
2. She has been playing the Founding Family game for at least 18 years (btw, that means she had Sarah while 17). For all she knows - whatever she finds might be something deliberately planted by the MC.
3. She might actually have something to do with it -
I mentioned a few pages back, that Hamilton's goals aren't bad, it's his methodology that's shit. Considering it's very doubtful Lily married for love... her view of the whole thing might be different. a few days of distress for a life-time of security and prosperity? Ain't that bad of a deal. Probably a lot better one then what she got as the third wife of Hamilton.
4. She might just not know how to find out about the picture. She might be good at social quips, looking good and being charming, but she might be dumb as a rock underneath it all.
And even if she wants to, could do it and finds out... so what? Anyone who imagines her ripping Hamilton a new asshole has watched way too many soap operas. Her entire life is dependent on the fact she is married to the guy since she was at least 16. Her daughter's well-being is dependent on it. What could she possibly do? Risk it all on a mad throw of the dice, hoping the MC is gonna become a major player? That's the field of the desperate and those who have nothing to lose. She is neither.
Like, I don't like her much, I think she is a pos. From what she has presented it seems she was just a gold-digging bimbo, that is batting to gain a higher position in the pecking order inside her husbands harem. But that doesn't mean I'm right.