I can say a word about some experiments I did with Peltier modules some time ago. I wanted to do more quantitative measurements before reporting but didn't find the time yet so here's already some qualitative feedback:
I used 4 Peletier modules (TEC1-12706) wired in series and hooked to 12V power supply. I intended to have the Peltier function at low intensity, this is where they are the most efficient, but it also means they don't transfer as much heat compared to having them run at nominal voltage. I had them sandwiched between two aluminum radiators, with some insulating polystyrene around to try to only have thermal transfer through the Peltiers. I added fans to the setup to maximize the heat exchange of the radiators. They were 24V fans hooked in parallel to the 12V supply so they didn't draw much and still were rotating fine.
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I tried to cool down a polystyrene box in which I cut a rectangle in the cover to put the whole "fans-radiator-Peltiers" sandwich in. Overall the insulation of the box must have been quite significantly lowered by doing this, but I went on. I tried a few scenarios, including cooling down the box air (without anything in) 10°C lower than ambient temp. It could not reach such a lower temp with the setup running at 12V, but more like -7°C compared to ambient temp. The whole setup with the fans was pulling around 1.2A, meaning you're burning 15W to keep the enclosure 7°C cooler (with my 16s battery in, that should not change the insulation of the box, just take longer to get to this temperature).
Warming up the enclosure 10°C above room temp was easier to achieve (by reversing the polarity of the power supply on the Peltiers), which is expected since they themselves produce a bit of heat while transferring some more through them. So I tried to put a thermostat that would maintain the enclosure between +5 and +10°C above room temperature. On one cycle, the setup took 7 minutes to bring the enclosure temp from +5°C to +10°C (with the Peltiers and fans ON), and then it took 20 minutes for the temp to go back down to +5°C (with Peltiers and fans OFF) before starting a new warming cycle. That means a duty cycle around 25%, so you burn on average a bit less than 5W to keep your enclosure between +5 and +10°C compared to room temp.
A good thing to do would have been to compare that with the consumption of simple resistive warming in the insulated box before I cut a large whole in the cover, but I didn't ?
. Still I think 5W is too much consumption for this contraption, and that the box cools much faster than it should if it were well insulated. One thing that I think we didn't take into account for this setup idea, is that during the time the Peltiers are not running, they behave like a big thermal bridge between the interior and exterior of the box (all the more with the radiators!), and largely weaken the insulation properties of the box. So thermal energy flows through very easily. When you put a resistor in the box without cutting it and putting heat transfer radiators on each side of the hole, the heat should be kept much more easily in the box. And I don't think that the slightly increased yield of Peltier modules compared to resistors is nearly enough to compensate the huge thermal leak.
So from all this I would keep two things: heating the battery in winter with a Peltier setup doesn't seem to be a good idea, resistive heating in a well insulated box should be better (although I didn't test it...). And for cooling, yes it can work, but it will suck quite some power if you want the temperature difference to be significant (here I got 7°C delta for 15W). Then again if it's summer and you have too much solar energy it can be fine.