One of the people on the CorsetMakers group on LJ suggested, among other things, Borax.  I’ve used Borax many times for spot-cleaning on corsets, but it hadn’t even crossed my mind to use it here.

Borax, a little water, and a toothbrush cleared up the darker spots and evened up the tone of the stain everywhere I used it, with hardly any scrubbing.  I must have been mistaken about the darker spots being true scorching.  It came up so easy that I think the darker places must have just taken on a much darker color because of the heat, rather than any actual damage to the fibers of the fabric.  I’m surprised and relieved.