[Spridgets] Tapered roller bearings

[derf]

OK, if the play goes away then it aint s'pension.
You can drill another hole in the spindle at around 90 degrees to the
hole that won't line up, probably not the best idea.
You could also turn (or grind) a few thou off of the nut.
If you must shim I would think the shim would go under the nut but
over the washer.
Did you try swapping the nuts?  Maybe a new nut?
You could try just a nice thin washer under the nut, something around
1/16" thick.
Cheers,
Derf