Hi Owen,
Many of us have had the same sort of problem too  
I'm not sure TTS is necessarily the best approach to improve the rendering of the white. Normal TTS figures of say, 1 rad, 200% will only really affect edges, which will help to define the overlapping petals.
You could try using a large radius, 10 to 16 and a low %, say 50%. That could give more contrast across the petals. Also try selecting the petal rather than use Neutrals
Another approach which I've tend to do more often is to use a curve. Put a dip in the curve near the top end which increases contrast in the brighter tones. You'll probably need to straighten the curve lower down so other tones are not affected.
See attached screen shot.
Terry