According to the MO Archives Veterans database, there were two William Wittens (I only checked that exact spelling- lots of ways it could be spelled):
William Witten Pvt Co K 15 MO Cav
enlisted 1 Nov 1863 Mt. Vernon MO
discharged 1 July 1865 Springfield MO
William Witten Cpl Co F,C,G 24 MO Inf
enlisted 18 Apr 1862 Springfield MO
mustered in at Arcadia MO
discharged 8 Dec 1864 St. Louis MO
I don't know if this will help you at all. If you need to search other spellings of the name, go to this website:
Be sure to type last name a comma and first name. However, the nice thing about this database is you can search by any three letters of the last name, or first name, and combine that with a unit number. But be careful. You should select Civil War from the drop down menu just below the man's name. And if he was in, say, the 1st MO Cav and you enter 1 for the unit, it will being up any unit with a 1 in it (10, 11, 12, etc.) which can be troublesome at times.
But it is a great database!