Looking at it again, I've realised that the minimum will be limited by the TL431 minimum voltage, plus the optocoupler LED voltage drop - likely 1.8V or more.
The LVM431 may work OK as far as that goes, with over a volt less drop?
The output voltage range will be halved, though. You should be able to compensate for that by reducing the value of R15, eg. 2K7 or a 4K7 in parallel with the 5K should get it back to somewhere near the full range.
Another option would be re-arrange the optocoupler drive circuit, so it can work at a rather lower voltage:
In essence, add a PNP transistor with emitter to positive, then an emitter - base resistor and a resistor from base to the TL430 / C5 junction.
Then connect the opto LED and its series resistor between the transistor collector and 0V.
That should give operation to at least a volt lower output, but is more likely to cause stability problems.
Try the LVM431 first?