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I don't think that this is going to work. With the majority of your trips being short and with the large mass of the transmission that would need to be heated, the engine would remain in open loop longer. Negating any fe returns.
Possible causing a loss of fe.
A engine block heater along with a silicon pad heater attached to a insulated transmission filled with the lightest approved hypoid gear oil would be best. Just put it on a timer so it's preheated before you leave in the morning. You could also put a small 110 volt ceramic heater on the floorboard inside the car to pre-warm and defrost the windows. This would help allow the engine to reach closed loop faster by not using the heater core to provide these functions.
If however, you want to heat the transmission quickly. Just attach a pair of welding leads to each end of the transmission case. Set the dial to 250 amps, turn it on and in no time your transmission will be fully preheated and ready to go...
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