Its been while since we used a three-way fridge on 12 volt mode, but here's my take.
Its likely in your case that the voltage dropped across the charge wire while supplying the current to run your fridge on DC. A significant amount of current constantly going to the refrigerator would not allow the voltage to get high enough to charge the battery.
IIRC A refer on 12v draws quite a bit of power. Also, outside temps matter as, on a hot day, the refrigerator will run almost constantly and that will draw more from your battery. Also, the combination of distance, small wire size, and corroded hot or ground lines anywhere leads to voltage drop.
If you have a combination of these things happening - small line size with corroded wires plus a heavy draw from the fridge, for example, than the battery cannot charge. 13.2Vdc is considered a "float" charge and 14.4Vdc is a "bulk" charge so 12.5Vdc is way too low.
So basically, unless you have enough volts to push the amps into the battery, the amps just won’t go in and you will show a low state of charge or even a dead battery.
We used to run on propane with our old pup's three-way fridge as much as we could.
DW, 3 Kids and our Goldens
2012 Shamrock 233S
2008 Toyota Sequoia 5.7L 4WD