Only adds a couple of ounces, tops. Even your high-rent tube-and-fin holds very little oil.
Be careful not to overfill. The cooler will drain back to the crankkcase after shutdown. If you add too much it will pump itself out the vent on startup.
Best way to determine exactly how much to add is to remove the cooler and lines, fill with water and dump their contents into a measuring cup, but I've measured a lot of coolers and you can be safe with adding a couple of ounces.
Nice to see that the "competition" now has an adapter that faces forward instead of aiming directly at the stock exhaust pipe......