I’m not entirely sure what would be needed to make your 03 trans similar to an 05 electronically. Keep in mind the 04.5-05 trans were the only year of the tow haul only with no third gear lock out like your 03 has. It also has a Throttle Valve motor on the side of the case which electronically controls the throttle valve. Your 03 trans doesn’t have this nor the wiring and electronics to support the TV motor. My 05 has the tow haul mode, I would much rather have 3rd gear lock out instead. Though a simple electronic box from BD fixes this issue.
For the temperature issue, more cooling capacity would be needed via an auxiliary cooler. I’ve been shopping Derale auxiliary coolers for my 05. I can fit their 15960 between my transfer case and rear of the cab under my truck. It’s their largest auxiliary trans cooler I have found with a stacked fin plate design.
What temperatures are you seeing at peak?
A deep pan gives you more fluid capacity. Meaning it will take a longer amount of time for the fluid to reach operating temperature. I run a PML deep pan mainly for draining fluid easily and a rigid machines flat sealing face. Yet, my pan gasket still leaks.
Glad your rebuild is working for you. A good converter makes a world of difference with these transmissions.