I mostly avoid cutting in to the sides that much. Why take chances?
However, it gets more "OK" the closer you are to the tips, the less bend that area takes, and you can get away with more of it on a more highly crowned stave. I learned this when I discovered that I could make bows from some woods like serviceberry, apple, hawthorn, and plum where the grain does complete barber-pole revolutions around the stave, as long as I started with a small diameter piece. This of course means that the width of the stave limits how much, as well.
But, I would absolutely have roughed out such a stave (so little deviation) and done the straightening at the same time I did the belly temper.