It'll be ludicrously expensive to built, and then you'll also have a ridiculous amount of heat to get rid of. The main challenge will be keeping that from melting the ice again (this, in general, is the underlying issue with TECs: they don't have any means of stopping heat from flowing back through them, so it's a bit like trying to bail out a leaking boat, and it gets worse the larger the temperature difference you're trying to achieve. TECs max out their efficiency at a mediocre COP at very low temperature differences, in comparison to compressor based systems which can achieve better COP with more than 10x the temperature difference)

Heat loss due to resistive heating of the thermoelectric elements is a factor.

I do some work using TECs for cooling optical sensors. Most of the industry has settled on a very small range of materials, such as BiTe, meaning that we're probably at a plateau until a more efficient material comes along. Some of my applications are battery powered, and cooling consumes a fair portion of the total power budget, so a more efficient cooler would be welcome.

Also, TECs are mechanically delicate. There's a laundry list of needs such as being able to form the material into the desired shapes, and make a good mechanical and electrical connection to it, which is usually done with some kind of low melting solder.