Hi Tim,
Good news on the PWM - that should help your resolution somewhat. It still seems like you are largely voltage driving the LEDs and reducing supply voltage is the last thing you want to do as it makes you more sensitive to the total LED turn on thresholds. Instead, try to keep the resistances as high as possible as this more closely approximates a current source. A true current sink/source is immune to voltage supply variations (within its compliance range) and doesn't require a supply regulator. Don't add cost if it doesn't add value.
Given the very low currents and the limited temperature range you may have to meet it may be possible to drop the op-amp in the first circuit I sent. It means the transistor won't turn on until the base is at about 0.55V but it should then follow it up until the base is at 5V. Feed your PWM via a simple RC filter to the base, set the PWM to near 100% and select an emitter resistor to give you the maximum brightness you require. Then tune your PWM to whatever light level you need. It's not as exact as the op-amp version as you have the offset to deal with and there will be some thermal effects as the base-emitter junction drops a couple of mV per degree C. It may be sufficient and it still wont require a regulator in the supply.

Best regards,
David