The TLC5916 datasheet seems to say that the minimum regulated LED current is 5 mA, but that’s painfully bright at, say, 12:08 in the early morning. Indeed, those 3-inch blue LED digits lit up the entire house from the living room… sometimes, a high-efficiency LED isn’t what you need.
This graph from the datasheet suggests that the current can be somewhat lower:
With that in mind, I replaced the 1 KΩ resistors with 3.9 KΩ parts.
The graph says the maximum current should be around 5 mA and, indeed, the formulas indicate 4.7 mA. The minimum current is a paltry 0.4 mA: lo and behold, the early morning illumination became bearable. After I put the LEDs behind some dark-gray polycarbonate, it’ll be just about perfect.
If it’s too dark, I can always solder another SMD resistor atop the 3.9 KΩ chips.
I figured out how to compute Rext somewhat more easily than the datasheet would have you believe and documented the process there.