I found this in a book called "Correct Misproununciations of Some South Carolina Names" by Claude and Irene Neuffer, University of South Carolina Press, 1983, p.3:
"ALE-li-WINE, AL-WINE The pronunciation of the name varies, even amoung cousins. Of German origin, the Alewine family lived in the upcountry Anderson and Abbeville area in the early half of the twentieth century and gave their name the three-syllable proununciation. In 1982 there were eight Alewine listings in the Greater Columbia telephone directory, some from the upcountry family and others more recent comers to the state, both pronunciations are represented."
I say ALE-WINE, like the two drinks!
But the most comon proununciations I have encountered is A-li-wine and Al-i-wine.