"White lithium" grease scares me, because it is a VERY general description, covering a large number of non-interchangeable products. One producer alone -- Fiske Brothers Refining -- makes well over a dozen different varieties and consistencies of white lithium grease. What's worse, some white lithium greases seemingly precipitate marble chunks; in my personal experience, Pennzoil 705 has been the most likely to petrifying, but there have been others.
I do use white grease for a fair number of light-lubrication uses -- usually Lubriplate 105 -- which is a soft calcium-soap-thickened grease. Still, I'm finding myself using Super Lube synthetic grease more and more often, especially for rolling-element bearings.
All that said, my inclination, if faced with your situation, would be to stay with oil lubrication for both the ways and ball screws