If you can't find a listing in a regimental history, try civilwardata.com (subscription site) which lists personnel under their "Regiment Personnel" listing. I searched under Leizer D. Pitts's name under "Personnel Directory" initially and it brought up a link to the Richmond "Johnson" VA Light Artillery Batty" which includes Clutter's.

Incidentally, it doesn't appear to have any further info on Leizer D Pitts himself, other than what you have already mentioned.

Regards - Anne