Oh my, there's another case of a "mechanic" who should probably keep his mouth shut.
With NO harm to my car, I've towed the caravan you see in the pic (taken on the Ernest Giles road) at near 130km/h. No problems. Used 17 LPHK which for my 4 cylinder diesel is pretty good considering the speed we were doing (1.8T van).
I always drive in overdrive. If we're climbing a steep climbs I'll drop the gears manually, and I'll engine brake on steep descents as well, but at all other times I keep it in top gear. We went to the National Navara Muster last year (was near Warwick in Qld) with a busted fluid coupling on the radiator fan and didn't have any difficulty.
We now have a 2.5T van and I'd do the same, no hesitation.
It's an automatic for a reason: it will automatically select the gear that it needs to for the current load and demands by the driver. Up steep hills (and really, I'm talking 12/13 gradients here not speed humps, try Cunninghams Gap) you could kick it back manually like I do, it does help, but ordinarily you should feel free to let the gearbox do its own thing.
Optimum RPM - should be about 300rpm less than mine (your optimum cruise RPM is 1700rpm, mine's 2000rpm). When hill climbing, I try to keep my RPM at around 2400-2500, so you might aim for 2100-2200 rpm. I aim to keep the engine load % when towing below 80%.
If you can't do that, then your car has a problem. The very first thing I would suggest is that you get the engine up to normal operating temperature, stop the vehicle, pop the bonnet release and turn off the motor. Open the bonnet immediately and try turning the cooling fan over by hand (be careful, EVERYTHING around it is bloody hot!). If you can turn it easily, your fluid coupling is shot and needs to be replaced = problem solved. It's the first thing I'd suspect.
That a problem happened after a particular service may be coincidental. Let's do a few things to try narrow down the possibilities - starting with the fan test. Oh, don't try stopping it with a rolled up newspaper - you could break a blade, and suddenly a free check of a component becomes an expensive repair job.