Surely the Disco is
designed for towing (unlike a Jag), so the oil cooler should be big enough.
The main thing that heats up the box on hills is constant gear changing. The recommendation is to change down manually on hills (and remember to change back up afterwards???). Manuals aren't always better - the 200TDi has a known fault which causes the manual box to lock in third after a lot of motorway-speed towing.
NO NO NO!!!!
What heats up automatic gearbox oils most is the normal slippage in the torque converter, or the BIG slip when the revs are under about 2,000 rpm. The recommendation to change down manually is BECAUSE this increases engine speed (torque converter rotates at that speed) specifically to try to keep the TC speed UP past the point where TC slippage develops. At 3,000 rpm for example, TC slippage is very low. The more the slip the more heat it makes. Worst case is stopped car, in gear, heaps of throttle: also known as "stall test". FIVE SECONDS of that is more than plenty. 30 seconds might have the oil so hot the smoke starts, and soon after that it goes on fire big time.
Only small amount of heat will be generated in the planetary gearsets, and a tiny amount in the brake bands and clutchpacks that create the gears. If any of these units FAIL they will slip, and that makes heat, but you won't be bothered by that as the car won't drive far anyway.
Even if you have a LOCK UP TORQUE CONVERTER, unless it is actually locked up the TC will be slipping and generating HEAT which goes into the transmission fluid. The 'Lock-Up TC only locks up after a certain road speed has been reached which may not be possible when towing a caravan. The heavier the load you are pulling, and the steeper the slope, and the hotter the day ... and the more ratty the kids are ... sounds like caravan towing to me .. and that oil gets Goddam hot. It could easily top TWO HUNDRED degrees CENTIGRADE, which is almost 400 real F degrees.
You need a BIG oil cooler added to the transmission (gearbox) cooling system, and if it were me I would ADD one that exchanges heat from the hot oil direct to the coolant water. Water is about two thousand times as effective at absorbing heat as air is. Leave the present cooler in the flow and add an oil-water cooler in series.
And if you have an electric fan, dump it in favour of the original LR fan whenever you are towing. That's my advice. Leccy fans can't shift nearly as much air as reliably as a bloody big fan driven direct by the engine, AND the real fan is always cooling the whole engine bay which the leccy fan does not. That is definitely worth doing as the turbo hot end and the first couple of feet of the downpipe may get red hot. A good constant draught of air will help to stop that heat from getting OTT.
CharlesY