In the bowling event (Part of Kali story) Moon already knows the MC name (Which she doesn't learns until after the 2nd bar event (After the training is over, where the MC introduces himself as Mr. Moon), there is also the possibility of her blowing the MC (Which then goes into conflict with the event where she invites the MC to her apartment and end ups blowing him for the first time).
What i mean about the bar is making it a part of the hotel (Not a buyable upgrade, but a free one) after one week or so of gameplay, that way the MC will at least know Moon before events involving her that make no sense with not meeting her (Pixel/MC taking Ally to the park, where they meet Moon) and the whole Kali thing.