While mulling over the DRO situation, I clamped the compound rest to the cross slide, backed the knob to the limit of the backlash, and poked feeler gauges into the opening:
The backlash turned out to be around 20 mil = 0.020 inch = 0.5 mm, which seemed excessive to me, so I fiddled around with the contents of the Big Box o’ Polypropylene Sheets (harvested from various clamshell retail packages), deployed the hollow punches, performed some deft scissors work, and made some shims:
Eventually, one of ’em offered a Good Enough combination of reduced backlash and E-Z turning to suffice for now. The proper solution involves facing off / rebuilding the fat metal washer on the right to put the bore at right angles to the bearing surfaces, but that’s another project.
The final backlash ended up around 4 mils, with a bit of drag due to the slightly irregular metal washer on the left preventing anything tighter. The cross slide knob also has a bit of backlash, but the thinnest sheets are a bit too thick.
Polypropylene isn’t the right plastic for a bearing, but it’s cheap, readily available, easily worked, and served as a bring-along project at Squidwrench…
The Smell of Molten Projects in the Morning 
At one point I purchased a tiny little needle thrust bearing to do exactly this for the cross-slide on my lathe, which is actually finished pretty well but I figured it’d be a nice addition. I have yet to actually install it, though. Still, something to consider. It’s definitely more than 0.020″, maybe 0.10″.
I could machine down the washers to give enough clearance and come out with a pretty good fit, but, in this case, the phrase “putting lipstick on a pig” springs, unbidden, to mind.