You'd be surprised just how much a steel rod will flex, even under light pressure. A 3/4 in. diameter, 36 in. length rod will easily flex .010 under finger pressure.
The .002 slop is troublesome. I'd suggest using adjustable linear bearings to set the preload on the bearings. I don't recall seeing any closed pillow blocks with this feature.
Check out some Thomson adjustable open pillow blocks.