[chromo]

"while I was detailing it, I found a small spot on the rear bumper, about an eighth inchin diameter, where it is starting to rust and peel. Otherwise, the bumper is immaculate."

Surprised no replies so I'll give it a shot.
You need to remove all the rust and seal the metal to prevent more rust.
If the plating is peeling then it's rusted under it surely so that loose plating needs to be removed to get to the metal under it to clean out the rust. To be sure you get it all you need to remove a bit of good plating also.
Not knowing exactly what you have I can only guess and mention what I have done before in similar cases. If you see 1/8" dia problem then go for about a 1/2-1" repair area (as a guess).

Chip and peel off the loose plating, but don't rip more loose! Sand the bare metal and feather the plated edges back aways with a medium to fine paper.
Clean area well before and after! Metal prep, acetone or other similar items to remove dirt, grease, oils, waxes etc..

Once area is cleaned up good, primer and paint to seal it just like you would any body part like a fender. Since it's an area not seen this should help protect from or slow further problems.

You can also get a Caswell brush plating kit for a low price and rechrome the area that way instead of painting it if you want. That would be better.
I have done this on a couple vehicles that I wanted to fix bad spots that were on the bumper front and seen, but the vehicle was not worth rechroming the whole bumper, just old work truck. I did not care about perfect I just wanted it a bit better, and it turned out pretty good.
With practice and a brush kit I made some other bumpers look far better also though not perfect and I stopped the rusting.