You don't really even need to center the stand. You just need to reference the rim to one of the two caliper arms (take your pick). Then flip the wheel over and see if the other side of the rim is the same distance from that same indicator. In fact, most of the time with "dual-sided" stands like the Park, the second arm just gets in the way so a lot of users jam a spare spoke wrench into the pivot on the unused arm to keep it as far away from the rim as possible (you can also remove it entirely, it's just one bolt I think). Otherwise the second arm is always bumping into the rim early on in the truing process when the dish and lateral true are still way off.

In fact, there's even a build method (described by Wheel Fanatyk in his blog, IIRC) that relies on keeping the dish way off to the drive side until late in the process. Having the wheel centered over the drive side hub flange is supposed to make radial truing much easier, plus keeps the DS tension relatively low for most of the process. Then you finish off the wheel by tensioning the NDS spokes and pulling the rim to the center. Or maybe it's the other way around. Anyway, I've never tried it but it sounds interesting.

A nipple driver with a usefully long, or adjustable, depth gauge should help keep the wheel relatively straight from the start and hopefully would minimize this problem. I've been eyeing the Problem Solvers one that Wheel Fanatyk sells (I also occasionally get close to impulse buying their tensiometer but so far have been able to stop myself ).