Finally got around to fixing this. Once again I'm reminded why I do this work myself. Turns out the 'professionals' who replaced my caliper 2 years ago after I bought the truck used the wrong part! Must have been one for an older LS gen 2 which uses a narrower bracket. My old rotor was worn thin enough that it didn't cause much of a problem until I replaced it.

So I got a new caliper and the bracket has far more clearance. I also tightened up the bearing adjuster just 20 degrees or so and it feels much better. Now the play is barely perceivable...just to the point where I can sort of feel a hint of it but not enough that it visibly moves or clunks when I rock the wheel. Everything still spins very easily so I believe I've found the sweet spot. Thanks for the help!