I took it out, swished it around and rearranged it every day for the next four days, toward the end of which it seemed not to be taking on any more color. After that, fears that it may have eventually started to putrefy drove me to take it out. The initial plan was to let it dry, rinse out any unfixed dye and bits of madder root, and put it back in the bath if the color wasn't satisfactory.
The chamois after drying but before rinsing.
I rinsed in about half a dozen laundry room sinkfuls of cold water without soap or detergent (per the advice of RAT members). After several rinses, the color of the water began to redden less, but lots of leather fibers started slipping.
Both after the initial dyebath and after the rinsing, I pressed as much water as I could out of the leather, then draped it over a box fan on low to dry, kneading and rearranging it every half hour to keep it soft.
The finished chamois is noticeably lighter and duller than before rinsing, though not quite as bad as it looks here (it actually has a bit more yellow than you see). Interestingly it seems that most of the slipped fibers during the rinse came from the shaggier side, which I take to be the flesh side. It's now much smoother and more washed-out in color than the shorter-napped side pictured above.
So, success? Well, for a given measure. The leather is certainly red. It's just not as rich or as even in color as we've come to expect. For all I know, this could just be what madder-red leather looked like in ancient times, but maybe there are factors that would result in a better color, such as larger vats or more frequent stirring.
One last note: The plastic bucket I used was also stained. I wonder, if I use it for future dyeing projects with other colors, will they pick up some pink from this one?