Hi Philip,
The Tennessee State Museum has a numebr of flags on display and even more in storage. Research appointments can be set up through curator Candace Adelson.
Union regiments were indeed issued two flags but the flag taken at Nashville was probably their National colors as the regimental flags more often than not, did not come with such fancy slogans as the flag of the 13th USCT did. This would typically be on the stripes at the lower end of the canton.
Greg Biggs