The Battle of Champion Hill took place on May 16, 1863. The date of Mrs. Lea's letter you cited as May 14, 1864, two days short of a full year after Champion Hill.

Brownsville does not show on my Mississippi State Highway map.

Enterprise is (and was) south of Meridian. Draw a circle ten miles in radius with Enterprise as the center and look for a small rural church or family cemetery at the distance of about ten miles from Enterprise in the northwest quadrant. "Ten miles this side of Enterprise" might put you near the small community of Chunky (think it was called Chunky Station in the 1860's). Another road leads out of Enterprise to the west - ten miles would put you halfway to the towns of Newton (aka Newton Station) or Hickory. I am assuming that Mr. Lea was traveling from the vicinity of Jackson eastward towards Meridian and then turned south towards Enterprise with your ancestor.