Losing a region of the display sounds like a whole row array driver chip is out rather than a backlight issue. I'd expect backlight would be intermittent, nothing, or a dim gradient starting from the edge where the CCFL is, and a bad "zebra strip" interconnect usually leaves one or more dead rows of pixels rather than a whole region. Shining a bright light on a LCD panel will usually at least reveal if there's an image even if the backlight is out, might need to look from multiple grazing angles.
I'm not finding a datasheet for this module but plugging in LMBFAT410G2CD turns up a couple vendors that at least have them up supposedly for sale, so that might be an option if the LCD actually died. I can only see part numbers on the actual row/column driver chips not the controller (likely the rectangular flatpack IC nearest the BS62LV256 SRAM), but it looks like a pretty typical sort of 20-something pin interface for a graphic LCD module so there might be other ones that are at least adaptable. From what I gather the geometry is 320x240 mono.
Worth checking the system power rails for sanity and hitting all the contacts with deoxit just to rule that stuff out, especially since you mentioned it wasn't responding to input (keypad doesn't seem to be integral to the LCD module from the look of it, suspect keys are just scanned by the 80C32). Just re-seating all the connectors may help, possible it got jostled during shipment and something was intermittent.
Looks like this balance uses a power brick of some sort, that might be something to check to make sure you're not giving it something like half (or double...) the line voltage it's expecting, and also just checking that it's producing the expected output voltage(s). Manual I found for what looks like this overall model class says it wants 12V AC or DC, so it sounds possible the power brick may just contain a transformer in at least some regions. With used gear they also may have provided a brick that isn't original and may be undervolting it close enough to make it work but not close enough to be stable, or overvolting it but not enough to instantly kill it.