Honestly if you want my opinion I think the beating came after she came home from Olive Garden. I think the likelihood that Lucas told her himself that the LL stopped by is pretty high in my opinion. I stated this once before but its been awhile so I'll say it again for those who may have missed but this is my theory on what took place after Olive Garden.
FLA, PB, or MSM(I think it was MSM but not 100%) stated that Lucas would say that Emily doesn't like him anymore. I think that Lucas being a child told Emily when she got back that the LL stopped by and how he didn't answer the door because in his little mind he thought it would make her proud of him. He told on himself without realizing it. I think he told her because he was so proud of himself for not answering. Poor little Lucas probably thought if he told her that she would start liking him again, that he would not get into trouble with her, that maybe the beatings would stop, and that she would be so happy that he did as she said. I think after Lucas told Emily the LL stopped by she IMO I think that after Lucas told on himself she snapped and killed him.
I say this because I know how children are and I post about this before but here is my personal experience with something similar that is a non abusive situation. Recently my little cousin who just turned 4 years old on the 13 of April has finally started pooping on the potty. One time his Daddy got disappointed with him because he pooped in his pull up and not on the potty. I was babysitting when he went on his own into the bathroom and went poop on the potty all by himself. He kept telling me Daddy was going to be so happy he popped on the potty and he couldn't wait to tell Daddy what he did. Again this was not an abusive situation with my baby cousin but as you can see kids get proud of themselves and they want to make their parents proud. Not that Emily is Lucas bio parent but I'm sure you know what I mean.
Anyways this has always been my theory about little Lucas. Now its just a matter of finding where the EG put him.