OK...so I think I'm gonna drop this game...decided to read through the comments and it seems like the third daughter (Rebecca) might be even worse than Lilly. I might be wrong here since I havent played too far into the game yet but it seems that she is manipulative and cunning which doesnt sit well with me considering her target for her schemes is her own family. So I'm gonna save myself the heartache here and just drop this one.
Lilly becomes easier to handle with time. You just have to keep in mind that when her father left, she was a total daddy's girl and it broke her heart. Her mother's marriage to Frank, in turn, has made the chances of her biological father returning one day even more unlikely. At the same time, she's probably worried that if she does open up to him, he'll one day disappear from her life just like her father did and he'll take Abby with him, so she'd lose them both.
But I can promise you, she'll get better. There is still a bit of drama to come with her, but the dust is settling and she is actually my favorite of the three sisters.
As for Becca, yes she is manipulative, but unlike her mother she doesn't do it out of spite but because she cares about her family (Abby, Frank, Lilli) and wants the three of them to stay together.
Frank's understanding of Lilli's situation is understandable: he knows her story well enough and knows that even if she explodes often enough, she is still very hurt. He will have had enough conversations about this with Shannon and Becca in the past. Apart from that, he knows that deep inside she has a good core, like her acceptance and protective instinct towards Abby.
I think Frank is a counterbalance to Shannon, so to speak: Shannon never really cared about Lilli coming to terms with the loss of her father, I think she will have either buried herself in her work or searched for answers at the bottom of a bottle of alcohol from time to time. Frank on the other hand was the exact opposite, even if she always pushed him away. Since we learn from conversations that the two had quite, not necessarily "good times" in the classical sense but a certain normality without excessive conflict.
Another point I can imagine is that in some situations she reminds him of his first wife Jessica, Abby's mother: Always in conflict with Shannon, an incredibly unlucky hand with men, general problems with birth parents but a good heart, unlike her mother.
I've had the thought in the past that if Jessica were still alive and she and Lilli got to know each other, she'd probably have some problems with her at first (Lilli does have some similarities to her mother at that age, and some tempered traits), but I think over time they'd get along, but I think they would get along with each other over time (even if it's just to piss Shannon off even more), when they realize that they are not so different and that she would be a kind of mother substitute for Lilli, which her biological mother is not really until the present of the plot.
Long story short: Give it a chance.