I do have a LPF on the contrast pin with a cap.
The R1 in your picture is built into the display (between Q1 and LED-) which is 6 ohms (two 12 ohms in parallel).
I think part of the issue maybe the way it is designed, I think it must have LED's in series. If I turn down the voltage to it, no current flows below 3.6V...
Here are some measurements:
4.5V = 81.7mA, 5V = 134.6mA (just the 6 ohm built in resistance) 10% voltage variation, 39.3% current difference
4.5V = 42.2mA, 5V = 68mA (6 ohm built in + 10 ohm external) 10% voltage variation, 37.9% current difference
8.1V = 40.7mA, 9V = 48.6mA (6 ohm built in + 100 ohm external) 10% voltage variation, 16% current difference
I only have 5V to work with, but this shows the issue. At 5V a 10% variation causes a larger current difference...