I have owned Dogs bigger than Deep South deer so no, absolutely not, you do not need tsx or ttsx or any other "premium" bullet to drop those tiny little deer. You don't even need them for the big 300+ pound Great Northern deer like in Canada.
I've used Hornady Interlock's and Interbonds for a long time and have yet to have one fail me as long as I put it in the right spot. And that is on Whitetail, Elk, Brown Bear, Moose, Caribou, Mountain Goat, American Antelope (speed goats), Bison, and Black Bear.
Used Rem CL's for years before I found the Hornady's and only had one failure due to bullet performance because I was just a little over speed at too close a range (bullet blew up on shoulder) still got the deer but had to trail it for close to a mile before I got another shot on the poor thing.
Cup and core bullets were killing Whitetail for many years before all these latest greatest "magical" "premium" bullets were ever dreamed up in the accountants heads. The only thing on this continent you should use a "premium" bullet on would be Elk or Brown Bear or Moose. Not saying that you HAVE to as I have never used them, but they do penetrate better so they give you better odds on that bullet getting into the boiler room. But thin skinned game such as Whitetail doesn't require it.