The casualties for the 2nd Texas Partisan Rangers were in the Dallas News (I think, dated 8-64) and gave a list of names for the Red River Campaign, no dates or locations provided. T.J. Brown is not on the list. That does not mean he was not a casualty but it does lower the odds. I doubt you will be able to find much in your search but keep trying. Local legends very often open doors and provide missing information. While it goes against what we know, the Red River was low and could be crossed in places so in theory some cavalry could have been ordered east for a while, etc. Harrison also had a couple of companies of Missouri cav attached to his brigade that seems to have been pretty active.
SB