Mihawk. I don't know what stage of the beginning you intended, but Mihawk should beat any version of Ichigo during the Soul Society arc. Bankai Ichigo might be a good fight and might have a chance but largely due to his speed. I'd still go with Mihawk over Ichigo at any point in SS arc. This is assuming reiatsu/haki are similar in this scenario. It gets complicated otherwise.
Well I guess he has a chance but like I said, I still see Mihawk taking it. The only way Ichigo would win is if he is much faster than Mihawk. I think he may be slightly faster but not enough to make the difference.