Ways back when I was a yute, I had one of them 650 Yammers that was basically a Triumph copy. I was living in San Diego where it is pretty hot too. It always ran really hot. I installed one of the Lockhart oil coolers, basically just like ours. It made a HUGE difference in engine temps. I would keep the oil cooler on.
The best test would be with bypassing as suggested, and installing an oil temp gauge to check the difference. I bet you will find it to run much cooler with the cooler. Hmmm, I bet that is why they call it a cooler maybe???
