Personally, I don't see any real alternative to how the problem that Lilly had with Frank could have been solved other than through Becca's manipulation.
Lilly was a total daddy's girl and therefore fixated on her father. When Trevor left, for reasons unknown, it broke her heart and she always hoped that he would come back one day, no matter how unrealistic it was. When Shannon married Frank, she realized that she would never have her old family back and projected her resentment, unjustifiably, onto Frank. In Lilith's Longing, we see towards the end that they didn't just have bad times, they also had some good times. We also know that Abby and Becca have always talked Lilly on the subject of her relationship with Frank, but she is incredibly stubborn. The only situation without her getting knocked up by some jerk and Frank getting to drive across the country to get her back because Shannon would have sent her away for sure was to put her in a situation where she was vulnerable and she had no other option but to open up to Frank (emotionally).
As for Becca: Through Rebecca's Reckoning, we know that Becca was more of a bookworm and introvert until she was 18. What pushed her into the starting blocks of her development was the realization that Frank, who she had feelings for for years, could be her biological father and that Shannon made her believe that Trevor was also her father. Then there's the fact that she didn't treat Abby very well, which is mentioned both in Shannon's bio in Single Again, and also mentioned by Abby that her relationship with Shannon was anything but good. I generally agree with her that it's healthiest for all concerned that Shannon is out of this family constellation.
Strictly speaking, I would even say that Becca and Lilly are quite similar. With the slight difference that Becca is subtle and Lilly is more of an “in your face” person (similar to her mother, except that Shannon is solely interested in Shannon.)