If it were mine....the left cover would be coming off. A little talked about secret in KLR land is that on occasion, even the upgraded, super-dooper, good-for-a-lifetime replacement springs.... break. Or come un-done. Either way, no tension is applied to the balancer lever to take up the slack when you attempt an adjustment. Eyes on will confirm or dismiss this possibility, and without an inspection port, pulling the cover is the easiest solution to me.
With the low miles you have, it is unlikely the rubberized balancer
sprockets have deteriorated, but only seeing will confirm this. Pulling the cover will confirm or dismiss the idea of chain dragging on the case, sprocket failure, poltergeist, etc.
I'm not always quick on the uptake, so bear with me.....you said when the engine comes up to normal operating temps, you hear the noise. Is this moving, sitting on the sidestand? Or both?
“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]