Yes, single compressor fridge freezers are fundamentally flawed when it comes to running at an ambient below the fridge thermostat setting.
True for most, but not all.
I have an over 30 year old Inglis "System 2000" fridge/freezer with a single compressor. It has a microprocessor based electronic control system. There are separate temperature sensors for the fridge compartment and freezer compartment (and also one on the evaporator coil for defrost feedback).
It has a single fan in the evaporator assembly that, when on, blows into the freezer. There's a solenoid controlled air valve that controls whether or not air from the freezer can move through a channel from the freezer to the fridge (from the fan pressurizing the freezer).
The way it keeps both sections at the required temperatures is:
- If the freezer is okay but the fridge is too warm: Open the air valve and turn on the just the fan (not the compressor) to cool the fridge from the cold in the freezer.
- If the freezer is too warm and the fridge is also too warm: Open the air valve and run the compressor and turn on the fan to cool both sections (like a common fridge/freezer).
- If the fridge is okay but the freezer is too warm. (The OP's situation): Close the air valve and run the compressor and fan to only cool the freezer.
(There's also a case where the compressor is running but the fan is off, for a short time after defrost, to cool the hot evaporator down before blowing into the freezer.)
The controller also reads the fridge door operated light switch, which allows it to turn off the fan when the fridge door is open, so it's not forcing cold air out. According to the manual, it also uses the count and duration of door opens to help decide when to run a defrost cycle.
Unfortunately, I'm not sure how this helps the OP. Maybe, in a way similar to my fridge, a thermostat circuit could be added that senses the freezer temperature and tricks the fridge sensor into thinking it's too warm when the freezer is.