I'm probably going to add an oil cooler to my diesel. But that's a different story.
If I was going to do it to the Enfield engine, I would not use the existing pumps, you are far too likely to cause serious problems (by overloading the pump, or destroying the pump if the cooler lines become blocked, ever).
Instead, I would tap in somewhere in the casing and use a 12v electric scavenge pump to move the oil through the cooler. That way there is no extra load on any mechanical system. If the pump fails, nothing is damaged. You can also include a thermal or pressure valve and a bypass in case of line clogging.