Moishe would be reasonable; however, the given name on this Hamburg departure list - left column, not far from top - following "Rabinowitsch" doesn't look like Moishe
http://goo.gl/s1yKvX. The Hamburg departure list indexers indexed the name as "Marscha." I think the German clerk who wrote the name would have recognized the Yiddish name Moishe. Maybe Moses pronounced his name "Mose-cha", and that's what is on the departure list.
A Russian speaker might be able to help with this.
Moses married his third wife, Jennie/Ida Heifetz, in Moscow after 1882. I might be able to locate their marriage record - but not on Ancestry, because I've already checked there. I am also going to try to locate the Lithuanian birth record for Moses.