What's like a heating element, but cold instead?

Authoritative cite.

I think this is the root of the problem. There’s a reason you can make an electric heater that uses small lines to put power in and get decent heat out, but can’t put cold out. That method of heating is not reversible.

Resistive heating

This. The trick you are looking for, I guess, is the “cleanest” way to transport that heat away.

This is an interesting approach to a solution. Basically, you want an ice box and a separate source of ice. Load ice in the box, get localized cooling. The cooling is done elsewhere and elsewhen. Gas expansion may be more effective than a block of ice melting, but it’s the same principle. Compress the gas into a canister, transport canister to site, use vapor expansion to get localized cooling. Depends on how big you want to cool, how cold you want to get, how fast you want to cool, and how long you want it to work.