Without a complete door seal, heat and moisture will enter the fridge from outside.
Heat: The fridge will attempt to cool your whole kitchen and house. Most french door fridges dispense their own heat via a fan the blows out to the front from the toe plate. From there the heat rises up in front of the fridge where it will get sucked in the gap between the doors. Your fridge will be running on a high duty cycle, perhaps very high, perhaps will not even be capable of maintaining proper (35 degree) fridge temperatures even at 100% duty cycle. Is that a bad thing? That's up to you. If you hate the fridge and don't mind accelerating its demise, and don't mind spending a lot on electricity ... sure. Unless it's really bad and your food spoils. You'll have to try it out.
Moisture: This is the bigger problem. All french door fridges are frost-free. They dehumidify the inside, and evaporate the water into the kitchen environment. Without the mullion your fridge will attempt to dehumidify the whole kitchen, and house. Initially the little evaporator underneath it will overflow and flood the kitchen floor. Eventually the system innards will freeze up with ice, and the fridge will stop cooling. Your food will spoil. If you're willing to manually defrost the fridge every so often, maybe every couple of days, maybe once a week ... you could live this way for a short while.
You say you have a thermometer and hygrometer in the fridge and everything seems to be normal. That is surprising .. I've had behavior similar to the above just from a slightly misaligned door seal. But maybe you're lucky, maybe there isn't much air exchange through the gap. It seems unlikely but if none of the above problems develop after a couple of weeks, you might be ok to run this way for a while just til you replace it.
Here's an option: If you're fed up with this fridge and just want to put it on life support til you get a new one, you could make your own mullion out of wood and screw/glue/silicone it to either door. The idea is to complete the door seal. You will have to open the two doors in fixed order because of the non-flapping flapper. But the fridge should otherwise continue to function normally. You say there's a wire in the original mullion? I don't know what that would do... you'll have to figure it out and perhaps bypass it if it has some kind of sensor or alarm function.
