Hardware is fucking hard. Peltier coolers are not nearly as trivial as they sound, they're wildly inefficient and get exponentially more inefficient as the temperature differential between the sides increases. To actually get the theoretical max duration out of the batteries you need to have thermal simulations and modeling to create the interfaces on each side such that the pelt isn't just sitting there freezing one side and melting the other while nothing happens to the payload. You need to have drive electronics that allow the pelt to run at a steady state instead of slamming it on and off randomly. You need a PID loop that's tuned for the thermal mass of the enclosure and payload. Finally, you need a ton of engineering on the enclosure and insulation such that it is volume optimized for the exact maximum duration that they're targeting.
Tl;dr the coolers are inefficient enough that you need to very heavily optimize your design if you intend on making this actually work for more that a few hours.