3.3V is too low for LEDs and an inverter to power a CCT backlight would require a considerable current.
Can't you just use a switching regulator to provide a higher voltage, or use a higher voltage supply and a regulator to power the 3.3V part of the circuit?
I'm pretty sure all LCD's require more than 3V to work - and sometimes there's some really odd bias voltages required for the graphical ones. A small stepup switcher is ~1cm on a side and is relatively cheap.
Then when I'm about to mess with it, I finally realize that the part needs a -8V bias signal, which also doubles as the contrast adjust. That and it is a 5V part. Too much hassle.
OLED's seem like a better tradeoff - supply a single high voltage ~12V, and since all the parts are relatively new, all the logic levels are sitting at 3V.