Another axiomization defines a person such that my friend is one person, but the drugs are making it difficult for him to express his desires

I think that is simply factually wrong, meaning, it's a false statement about your friends brain.

One axiomization says I should consider how happy he will be with an outcome, and I believe he'll be happier once he's sober and capable of taking care of himself. Another axiomization says I should consider how much he wants a course of action, and I believe he'll be angry at my trying to run his life.

I think it comes down to this: you want your friend sober and happy, but your friends preferences and actions work against those values. The question is what kind of influence on him is allowed.