" I cant find the breather circuit diagram on the web to look for possibilities. "
Remove the petrol tank, avoiding breaking the plastic fuel pipe connection.
There are 2 pipes ( on the right behind the block ) coming from the crankcase near the clutch .
They both go straight up into the air box.
If you remove the air box end of the pipes ( easy ), you can look down them & will soon see if they are full of gunk.
On my old 2010 K1300 they were pretty clear after 50k miles.
But on Matts 2015 K1300 replacement with 35k miles, they were pretty well blocked, & Matt had just had a new cover gasket due to a small leak, & was still leaking slightly at the back.
My old K never did short trips, usually at least 100 miles each trip.
Matt used to use his ( now mine ) to get to work, so it had a lot of shortish ( 20 miles I believe ) trips which cause a lot more condensation in the oil.
Hence a build-up of mayonnaise oil gunge in the breather pipes.
No more leaks from the cover gasket after that.