I haven't found a military unit for him as of yet, but John Fletcher Wyche was definitely an important man in the Iberia Parish area. Portions of the parish bordered his property. John Fletcher Wyche also was listed as a cotton agent with a business in New Orleans.
I would tend to think that the fact he was so well established would have made him a candidate for having service in another war or that the title Major was simply honorary. I found that Wyche was born around 1835 and had a son born in 1869 to him and wife Mary P. Since he was a large landowner, you might peruse the records of Iberia Parish for a succession. Those documents may shine a little light on any military history.