I think misty cloud is right.
Although the liver is a regenerative tissue and living lobe transfer is possible and i believe has been done, I don't think it is a generally approved procedure unlike kidney or bone marrow although i think it it done sometimes.
I can only imagine the reason is down to risk to the donor or a low success rate for the recipient.
I imagne that even with a living donor, if in a trust that allowed it, the recipient would have to have had a certain period of time clean of Alcohol to be considered as it would affect their general health, also it isn't a full transplant so they'd need the liver they were attaching the healthy tissue to, to be as health as possible. Also, the need the the recipient, if it is from Alcohol Abuse, to completely understand what they are getting and not think "great new liver, I can start drinking fresh".
The cast majority of alk transplants will still come from cadaver transplants which means absolutely no link between the donor and recipient and no queue jumping, even if a close relative.