Since General Robert E. Lee was famous, his family genealogy has been documented fairly extensively by some historians. Since the relationship mentioned in the letter is described as distant, you may want to start by tracking the family members of both Robert E. Lee and his wife back for a few generations, and also forward at least to the early 1880s to look for possible connections to your wife's Bradshaw relatives who were alive during that era. Since the letter was written in 1886, that should help narrow the time frame a bit.
It also appears from your question that you have tracked some of your wife's Bradshaw ancestors from that time frame already, so that should be helpful.
There were some efforts by people during the years following the Civil War to document their family relationships to prominent individuals involved in the conflict. There should be some library reference sources available specifically focusing on that era concerning the genealogy of prominent figures in the Civil War. Federal census records might also be helpful.