Yes Sir,
Take it when you can get it.
But if you can plan it out for for Winter, the sap will be down and you get an extra full Late Wood Growth Ring to work with...
Nice thing about Mulberry, you can Peel the bark, with some effort, with a hatchet or strong knife.
It will come off in big thick chunks if you peel it fresh.
Wether you take the sap wood off or not, Seal the Heck out of the back and ends.
Taken in the winter Osage will tend to check less even if you leave the sapwood on.
But the best way is to take off the sap wood as well.
Then seal the Heck outa it...
-gus
-gus