I'm just going by memory on my FP430 here... (I don't recall you said what model you have)
I recall behind the panel on the fridge is a row of 7 or 8 screw terminals mounted right on the end of the power supply unit. The top 4 terminals (two blue wires, two brown wires) is the common 12 VDC circuit (and is also where the supply outputs it's 12 VDC as well). One set of wires (brn & blu) is from the 12V IN jack, the other set goes to the compressor controller.
Maybe one of the 12V jack wire screws is loose? (these screws being loose I know had been mentioned in the other threads) Or the common connection between the two blue wires or the two brown wires down on the supply's board was lost somehow? (check for continuity between the two blues, and again between the browns) Or maybe a wire came detached from the 12V jack itself? (just throwing out some possibilities here)
As a last resort, you could try applying 12 volts to the terminals where you measured 12.5V when the fridge was plugged into 115V (brn is +, blue -) and I bet it would start up and run (running it through your inverter (12V > 115 V > 12V again) is extremely inefficient).