Thanks.
I didn't say but I sort of tested that too.
I ran the arduino off 4 partially depleted AA's to see how it performed and due to the voltage drop the current through the display dropped by 0.5mA and it got a little dimmer but still seemed to be going strong.
The rest of the project should run on a supply down to 3.8v without issue but probably not the LCD.
In that respect I still have some thinking to do.