Some thoughts...
- This thing consumes the best part of 2W. With an ambient temperature of -5C, it ain't gonna be -5C inside the case! If the upper ambient temperature spec allows, a sealed case will keep all of this heat inside nicely, especially if the case is non-metallic. If you can place the voltage regulators near the display, so much the better...
- I would be highly untrusting of the freezer spray plus thermocouple experiment. You'll get a whole range of random temperatures, depending on the thermal mass of the items covered in frost, and the thermocouple will only be in vague contact with part of it. Put the thing in the freezer for a while, monitoring the temperature with a thermocouple thermally bonded to the glass.
- If the LCD really doesn't work, given the above, then it's the wrong technology and heating the damn thing up is just crazy - every way you look at it!

Sorry! It
might be reasonable if it's a custom LCD with annunciators or similar (like the German telephone), but here it's just text. Other than code (trivial excuse!), why is OLED out? It won't have an appreciable lag at low temperature - it's a diode!
- Alternatively, VFD would work, and they are available in 16by2 compatible versions (meaning no code changes):
http://www.noritake-itron.com/subpages/productse/vfmod7000.htm - e.g. GU112x16G-7806A - I've used these a lot, and they have no problems at low temperatures. They take more current than an LCD (without backlight) or O-LED, but no-doubt less that you're proposing to waste with a heater. And they can be switched off by software when not needed.
- LCDs are commonly used in cars, and they work fine below freezing. Yes, they might be slightly "laggy" before the interior heater starts to work, but what's the big deal? After all, they are only telling you non-critical information like the time and radio station you are tuned to. Actually, why is the lag a problem in this application - you're not trying to scroll text or anything? Can a rethink of the UI contribute towards an acceptable compromise?
- An old-school incandescent backlight might be a "nicer" solution, if you can work out the mechanics and optics.
- What if your heater system goes wrong?
https://www.eevblog.com/forum/microcontrollers/lcd-display-fault-any-ideas-what-would-cause-this/Good luck, whatever you go with!
Mark