Arduino/ESP32 to intercept and "spoof" thermocouple signal, is it doable?

I have very little knowledge in this field so please bear with me.

What I'm hoping to accomplish is the following: intercepting and, if needed, spoof the signal from a thermocouple (currently unknown type, probably type K) measuring outside temperature in a FTX system.

The reason is that the controller board in the FTX unit has very limited capabilities and is hard coded to always activate heat regeneration when outside temperature is below 15°C / 59°F, regardless of supply or exhaust air temperature. Regeneration is also only on (100%) or off (0%) so summer nights are a nightmare.

So my idea is simply to intercept the signal from the thermocouple and spoof it to trick the controller board when needed. I want to take the input from both the outside temperature thermocouple *and* exhaust air, and then be able to set IF statements like:

IF Exhaust air > 20°C/68°F AND outside temperature <15°C/59°F SEND spoofed outside temperature value of >15°C/59°F to control board (kill heat regen)

IF Exhaust air < 20°C/68°F AND outside temperature <15°C/59°F SEND unspoofed value to control board (activate heat regen)

So basically I *think* I would need two thermocouple inputs and one very accurate voltage (mV?) output from an Arduino or ESP8266/32, with suitable code of course.

Is this something that would be doable? The thermocouple amplifiers seem simple enough, but I'm not sure what I would need for the mV output. Something like an external DAC? From a quick search I found components such as the MCP4821 but have no idea if it's suitable.