Oversteer, your altruism is admirable, to lose a chunk of your own liver for a relative is no small sacrifice.
Compatibility test first then if suitable:
From the perspective of the medics, they will be aware of the difficulty which confronts JWs when it would normally be recommended to use blood in a procedure. Surely a part liver transplant would demand blood. The doctors also know that when push comes to shove, many if not most JWs will take the transfusion on the understanding that the doctors knowing the patient's predicament will not let on to the relatives that a transfusion has taken place. This usually applies in the UK, it might be worth knowing if it is the case for your sibling's doctors.
The first issue is the person's life weighed against their compliance with what they have been led to believe is "God's law".
Second, whereas you wish to help, your own anti JW position on blood will be exposed which will have hefty social consequences .
Third if you permit your donation to your sibling he/she will see it as being able to live only by complicity with sinful anti-JW thinking.
Your sibling's instinct is to lose life rather than get mixed up with a non JW style resolution. Whereas blood is the safest and quickest means of recovery from serious operations. You can give them a tangible hope of a future life, the Watchtower's hopes have never been realized.
Can you explain the whole background of obedience to this JW org has all along been a waste of life? Probably not... and in this particular life and death situation the JW instinct, after years of indoctrination, is for them to simply conform to the demands of the cult.
I imagine that unless you can convince your sibling of the bigger picture, you are likely to be keeping the whole of your own liver.
Of course you are the one who is in the unenviable position to make up your mind on the matter but your sib is under great pressure to perform as an obedient JW. A person who has been trained to give all to the WTBTS and people when under such immense pressures are less likely to make rational decisions.
So might the answer be to have a further calm long talk with your bro/sis to get them to see reason and if unsuccessful from your position, you can still retreat knowing that you gave him or her your best?
It must be in their interest to listen to you and surely they would agree to keep silent on the matter of your disagreement with the WT on account of your willingness to help?
How sad it is that irrational thinking puts people into this dangerous medical dilemma in the first place, however I wish you the best, whatever choice is made.