There was an old Hitachi data book that described a
transistor circuit to do such a thing. In a nutshell,
the secret is to track the temperature coefficient
of the LCD display. A -12mv/C requirement for
the LCD can be met with a -2mv/C be transistor
junction multiplied by a gain of 6. Its not trivial
to design and prototype - but in manufacture, it is
very cheap - which is what you asked for.

You probably still need the manual pot, although
the Hitachi app note suggested even this could go.

Knowing the LCD temp coeff is the first step.
I don't have the Hitachi controller data book handy -
but perhaps I have given you enough info, here,
anyway. Good Luck.