Powering a 12V misting pump using batteries?
Two simple solutions. One. If your balcony has a light bulb socket, use a socket to outlet adapter.

The second requires running power. At 48W, you could use a higher voltage DC supply, like 48V 1.5 Amps (to make up for power loss) and use a local dc to dc step down converter. Just get the right gauge cable for the round trip distance with minimal power loss. At this point, you are essentially just using an extension cord with additionally circuitry. A regular extension cord would be easier.