So I tried something a little different. I used all of the default advanced inputs and I then scaled all the base mesh cell quantities by 9/8. So basically, it shrank and shifted the base cells every so slightly and the result looks pretty good for refinement level 2.
My suspicion is that when there is an asymmetry in the base cells across the cylinder, it is more difficult to “fit” the layering in without changing the cell structure significantly. By shrinking/shifting the cells every so slightly, the fit is much better.