The fact that it drives fine when cold indicates the TCU and the PTCM with the wiring harness is okay, but please look through the troubleshooting pages in the service manual.

However, there is no definitive answer to your question. When the transmission and fluid is cold it may be able to put enough pressure on the clutch packs and they work Okay, but as the fluid gets hot it can't maintain enough pressure on those clutch/metal interface plates and so it starts slipping. The same reasoning applies to the solenoids; the heat is causing the solenoids on the conductor plate to malfunction and those solenoids control the clutch packs shift points.

The clutch packs consist of fiction plates with smooth steel plates between them. The number of plates varies between gears and I think the SRT6 transmission has more plates than the Limited models. Looking at your picture with very black fluid those fiction plates and steel plates were slipping to the point of causing so much heat the fluid burned. The friction plates may be damaged to the point they must be replaced. Without opening the transmission it can not be actually determined how bad they are. You can try to install a new conductor plate (be sure it comes with new solenoids) add new fluid, new filter and the transmission may work okay. You can only give it a try and hope for the best.