I'm no expert on LCDs but when you say "battery goes down" are you talking a few mins before it goes flat, or a "slow death" throughout the day?
I ask because it may be that the logic in the radio is that the vox is the prime reason for having the radio, so keep that alive as long as possible at the expense of the "anciliary" stuff.
Either that or it's gubbed!
I take it you have proof positive it still TX/RX OK when the screen is dead? I ask because if you can't use it (change channel etc) because you can't see the screen it doesn't sound "perfectly useable" to me.
Worst case you replace it and carry it as a spare. If T/RX works and it's got enough charge to keep the screen alive for a few hours it will probably work fine as a backup set.