I think it is a matter of personal preference but I like the love path better. It seems like what she needs more because she doesn't lack discipline but rather affection. She obeys with both the carrot and the stick. Alice is the one who really needs some stick time.
Agree to disagree on that one yeh she is being a bitch but she is the older vision of Lily or what Lilly could of been in a few years time.
Like Lilly she was starved of her father's time and affection only were Lilly blamed her Eldest sister thinking she was stealing him from her.
Alice actually put the blame were it belonged on Daddy himself and has decided for her own self preservation if he dosen't want to be with her she doesn't want or care to be near him.
And the best defence she has is Anger at him so beating her with a stick won't get you anywere except despised and feared.
I could be wong I often am but thats my take on Alice.
I'm personally of the opinion the best he can do is to ignore her anger, rebuke her rudeness and keep making her have Daddy, Daughter time.
Your suggestion might work I'm no expert and of course my way is just my personal opinion.