I suppose a faulty temperature sensor is the obvious conclusion, but it isn’t a common fault. Unlike on the 5-speed models there’s no external wiring or connectors between the sensor and the ECU which could also cause issues. The 6-speed has the ‘tracks’ built directly into the electronic module, so I can only guess that the sensor itself has a fault.
The electronic module is coded to the vehicle so, unless you can get your existing one fixed, a replacement would require re-coding.
The radiator and oil cooler issues were on the early 5-speed L322s.
When you say that you “had it hooked up while driving to check temps” do you mean monitoring the fluid temperature via diagnostics (using the transmission’s sensor) or by some independent method?
A possible reason the fluid might overheat is because the torque converter lock-up clutch isn’t functioning correctly but that certainly wouldn’t cause the message to be displayed on start up when cold.
Can you get the codes read in case there’s some other electrical/electronic issue (battery voltage, etc.)?