I’d be taking the vehicle to a transmission specialist (as opposed to a general workshop). A good transmission expert will be able to tell you a lot about the condition of your gearbox by giving it a once over. That will include an examination of the level and condition of the fluid as well as an inspection of the contents of the transmission filter which can offer up an enormous amount of detail on what’s happening inside the gearbox.

When was the last time the transmission was serviced? The fluid in these units is a bit like the oil in your engine; left unchanged for too long, it will degrade and won’t be able to do its job which, in the case of an automatic transmission, includes both lubricating the unit as well as providing the actual driving force to turn your car’s wheels. If your problem was intermittent or was solved (even temporarily) by switching the engine off and back on again, then I’d be inclined to suspect an electronic or sensor problem. But since the problem only happens when the transmission gets hot, it does sound potentially mechanical in nature.

It could be a problem with the diesel pump, it might be a problem with the turbocharger, or it could even be an internal problem with the engine itself. Get it to a diesel mechanic for a proper diagnosis.

I wouldn't reconnect it. If the aftermarket cooler is designed to operate as a stand-alone one, and it's doing the job, then I would leave well enough alone.