Morning Bob. Let me throw out a "better late than never" welcome to the forum.
The KLR 600 nor the 650 are especially known for blowing head gaskets or cracking heads. While we do have members capable of getting an anvil to leak coolant, the bike itself doesn't have a huge history of this, like say, a 1996 Ford Taurus. While the two bikes, the 600 and the 650, don't share many parts with each other, the basic designs are very similar, neither design causing dealers or parts suppliers to stock heads and head gaskets like they would spark plugs.
The "clean spot" you found may very well of been the place coolant was escaping while under pressure. It only takes a hole large enough for fluid to seep through to contaminate your oil. The other possibility in the head area to allow coolant into the oil could be a cracked head. A machine shop that does head work should be able to test the head for you, and tidy things up while they have it. I have had a number of what I thought to be good heads found to be cracked once they got to the machine shop. I should point out, this has never occurred for me with a Kawasaki head.
There IS another place where there have been some coolant-mixing-with-the-oil problems....the water pump seal. Kawasaki Part # 49063. There are two seals at the water pump, the first being the oil seal. This one prevents antifreeze from contaminating the oil / oil contaminating the coolant. Next in line is a coolant seal, that prevents coolant from leaking out of the weep hole and / or into the oil. This coolant seal is what is called the "mechanical seal". If all is good in your engine's head area, I would look into the oil seal at the water pump.
Please keep us posted with your progress. If you can identify the problem and come up with a good solution, we might just sticky this thread, keeping it for the benefit of future generations of KLR riders, and placing you in the KLR Guru Hall of Fame. You get your own cape and dressing room, and two or three eunuchs to tend to your harem. Document everything!
“many a trip continues long after movement in time and space have ceased”- Steinbeck, [I]"Travels with Charlie"
[FONT="Century Gothic"][I]Sometimes your only available transportation is a leap of faith[/I] [/FONT]