So, I got to thinking that running back and forth to the store to pick up dry ice to cool the hot NUBM44 was getting a bit old. Paying high school kids to wave a hand fan at the diode proved not only inefficient but expensive.
While what I am about to propose is also inefficient, I wonder if it would work. One would have one set of batteries for the diode and another for cooling. Perhaps putting the 44 into an aluminum radial finned heatsink with one of these in between the sink and the module.
Otherwise, we are going to have to use mind over matter and just keep thinking cool!
You have written that TECs are only 10% efficient.
I just have bought the so called "Beverage cooler&warmer" from ali and checked it. It cools 300g of wáter put inside of the can at rate of half ºC pro minute from rt to about 2-3ºC (that takes 1h but at rt cooling starts at rate of 1/2 degree pro min).
Knowing that 1 calorie (needed to cool 1g of wáter at 1ºC) = 4.2 Ws, it makes 10.5W of heat extraction while energy consumption (as was given in specs) is 12Vx3A = 36W.
So the TEC in it must be 29% efficient (if we only care about wasted energy).
I took the thing apart - the TEC inside was not bigger than ones purchased from ali before.
And if something is cooled from temperaures higher than rt to rt, the efficiency should be even higher, should not it?