I seem to have been unclear, a skinwalker can control himself as a wolf and can effect the change at anytime, he is not forced to be a wolf on the full moon.

A werewolf is an uncontrolled killing machine during those three nights, the full moon and the day before and after.

A beastman cannot both transform in the full moon and control himself during the time. Either he has control and changes whenever he likes or he doesn't have control and changes only on the three nights of fullest moon.

Now as to the bsckstory, was this "stranger" a demon or a vampire? I'm guessing cause of the horns he was a demon. This being the case he wouldn't fear a werewolf or a skinwalker, still not sure what Alex was. Demons regard themselves as the dominant lifeform considering humans on the same level as cows or horses to be used and discarded. They consider vampires, werewolves etc even below humans because these monsters were born from humanity no matter how litte they might resemble it.

While werewolves may bite someone being unable to control themselves skinwalkers only turn someone if they have a reason.

This would be my advice, write in some sort of history between Alex and the demon, make it so for whatever reason Alex was hunting this demon, probably for revenge for something or other. Give the demon a reason to fear Alex. Maybe they've battled before and the demon barely escaped with its life and perhaps it is using the blood sacrafices to regenerate itself, I'll leave the details up to you.

My second piece of advice on your character being turned is to make it so he was mortally wounded on the alter before Alex arrived after killing/exorcizing the demon the only way to save your character was to give him thd strength of a skinwalker in order to heal.