We bought a 2011 Navara Dual Cab STX diesel automatic that we often use to tow a 2.8-tonne caravan. We were assured several times by the dealer that an external cooler would not be necessary, but following your comments in Carsguide I wonder if that's correct. I prefer peace of mind, rather that a surprise along the way.
Fitting an external cooler would do no harm, it can only help. The other thing to keep in mind when towing is not to let the gearbox lug in overdrive, but change down when the road demands it.
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