Low voltage LCD glass displays
You should be driving the common in sync with the segments, either in phase or 180° out of phase (depending on the on/off state of the segment). 

The DC content of your drive waveform should not exceed some tens of mV or you will damage the display permanently. If you are driving your (static) display with single ended port pins you will get less contrast and will damage the display eventually. To minimize the DC content you want the out-of-phase waveform to be very close to 50% on off ratio. You can work out the permissible error in time from the frequency (which should be in the range 32-100Hz for this display). 

5V is required for Lumex displays that operate up to 80°C, 3V is okay for 60°C. With your 3V supply (and properly driving the common) you will be driving the display with 6V RMS so it should be fine. 



It's quite possible to source LCD displays that fit your requirement, in fact custom displays are more common than standard ones. There is an NRE cost and usually a minimum order quantity. You get to specify the type of drive (static or the more common multiplexed), pick a fluid temperature range, a viewing angle, glass size, type of termination (zebra strip, FPC or pins) and so on. Figure on a starting cost in the low thousands of dollars for an ordinary display.