A bypassed heater core would not affect the electronic climate control system. I'd assume the problem is with the HVAC Programmer (which is attached to the plastic cover over the heater core), the climate control head, or the wiring between the two. If a sensor or actuator (perhaps the ambient air temperature sensor, or the air mix actuator) goes bad, the outside temperature will blink upon starting the car. In this case, it is larger problem.
My money is on the climate control panel. It'* the easier of the two (vs. the programmer) to replace.