The fuel temp does not affect the operation at all. Only oil temp. And by too cold RR mean around zero deg C.
I think we are saying the same thing with different words. I agree the fuel temperature has no direct effect on the bypass. However, the continuous flow of cold fuel would grossly overcool the oil unless the bypass diverted oil around the FOHE to maintain the bulk oil temperature.
All the fuel flow always goes through the FOHE, but at low fuel temperatures, most of the oil goes around the FOHE via the bypass.
Sooty