I feel like there must be a classical result answering this question (or easily modified to do so) but a quick flip through Soare didn't produce anything so rather than waste time I figured I'd just ask.
Given incomplete r.e.sets $C \nleq_T B$ must there exist a low r.e. set $A <_T C$ with $A \nleq_T B$?
I'm guessing the answer is no and somehow you can put together Robinson low splitting with one of the non-splitting/non-bounding theorems to show this but I'm not seeing it right away.
EDIT: Great answer to the original question below (thanks Ted!) but I realized I should have specified that I wanted the set produced uniformly for the application I wanted.