It looks like most of the voltage adjustment comes from the wandering reference, no wonder you said RV1 doesn't have much effect.
Assuming your Q16 path works okay, comparing the two diagrams, V(R42) dropped (6.22-6.56)/6.56=-5.18%, almost the same as the LED string voltage drop (11.4-12)/12=-5%, which means the 2SK246 ccs didn't have much effect at all. To give you a data point, my mocked-up 2SK246 ccs changes V(22k) from 6.5496V to 6.5446V (less than -0.1%) when the supply changes from 12V to 11.4V.
Your LED string seems not great, either. When the current drops from 703uA to 663uA (-5.7%, ignoring the Q16 base current), the LED string voltage drops -5% as well. Not much of regulation, isn't it? I grabbed a random red LED from my stash and in similar situations its forward voltage changes from 1.7914V to 1.7881V, only -0.18%. Your LED probably has 700uA right around the knee on its IV curve. It may not be a bad part, just not suitable for this location.
The above assumes the Q16 and the rest of the battery circuit works okay. Oh well, let's fix one problem at a time.