Great success!
I got a screw made for the spare tailstock spindle, so now when I had a backup available I decided to go ahead.
Armed with my $20 reamer from eBay and a few beers, I started out by turning a long test bar to see if the tailstock was centered. To get the best possible accuracy, I used the spare spindle. After some minor adjustments I was down to .0005 difference, which translates to a .00025 offset. Just about as good as you can get it.
With that done, I stuck the reamer in the chuck, stuck a live center in the end to make sure it was straight in the chuck, and tightened it up real good. I installed the old spindle and was ready to go... or so I thought. The spindle is a short MT2 and the reamer bottomed out, so my initial plan to use the screw to feed it went down the drain right there.
After another beer I decided to simply pull the spindle out until the reamer cleared, lock it with the spindle lock screw and then push the whole tailstock assembly toward the reamer. It kinda worked but it wouldn't cut, and after checking the reamer I found that it was dull. After some bad language and yet another beer, I started stoning the reamer. It didn't need very much, and after 15 minutes of stoning it was cutting like it should. I used plenty of cutting oil and went slowly, reaming, cleaning and reaming again, and after the 10th pass or so it looked really nice. I checked it by inserting an MT2 shank, and it stuck in place just like it should.
I cleaned, lubed, reinstalled and now it was time for the moment of truth: I put the test bar in the chuck, centered it on the live center, put a test indicator on the bar and read the same .0005 as before the operation! This means that the reaming was a success, despite using a pretty crude method and hand stoning the reamer. I celebrated this by having a beer, and I think I'll have another one after posting this!