I'm not convinced that Tie-ring Enfields were a staple in the TMD. Have a couple been found? Perhaps, but their origin is likely the Cis-Mississippi. As for French triangular base bullets, they are found on rare occasions in Arkansas, primarily in Confederate camps from 1864. Not sure what their frequency is in LA, TX, MO, or the former Indian Territory. While they certainly were used in the TMD, I don't consider them a "true" TMD bullet.
To me a "true" TMD bullet is one that is rarely, if ever, found in significant quantities east of the Mississippi. Such bullets include the MTE, 2-ring "Mississippi" (we need to come to a consensus on a better name for this one), "Arkansas Hog," "Sterling Price," "Cosmo Type II" (need a consensus on a better name for this one as well), "Tom Green," "Baby Belgian," "Baby Austrian," "Fagan," "Four-ringer," etc.