If one were to put an oil cooler through a dishwasher, how much soap and water would you expect to actually enter the cooling passages? Worries about entrained water and detergent corrosion are exaggerated and baseless. Blow the passages clear with some compressed air and get on with the installation.
By the way, unless you operate in an extremely hot climate or subject the car to endurance tests, isn't the oil cooler an unnecessary piece that actually prevents the oil from ever reaching optimal operating temperature. For example, with a 300 SL in road service, most owners bypass the oil cooler so that water and oil temperatures stabilize at approximately 180 degrees. What oil temperature would you expect a Pagoda to reach with an operational cooler; probably nowhere high enough to serve efficiently. You might best be served to replace the cooler for appearances, but bypass it to raise your oil temperature.