Aaaah, mulberry my most hated/loved woods! Loved because you can chase a ring, it is beautiful and strong. Hated because every stave I ever had was plumb infested with clusters of pin knots and I had limited experience.
Take your time chasing a flawless ring and treat like you would osage. Just give yourself an extra quarter to three-eighths inch width and an inch or two extra length. Dry mulberry responds to dry heat or steam. I would recommend you double coat it with shellac if you are going to steam dry mulberry, though. You can later scrape off the shellac with a sharp knife, sand it off with sandpaper, or wash it off with an alcohol soaked rag. But the shellac will be a little insurance against the light checking that can come with steam and dry wood.
The hackberry stave with the bark on looks as if it will have some character. How in love with it are you? Hmmmm? Hmmm?