They make heat tracing that is waterproof. It's basically a cord that heats up when power is put thru it, you could run some of that around the bucket to keep it warm. One thing I use for my planted in the ground trees is, I get a string of christmas tree lights, not the led but the bulb types and wrap them with those then pull a blanket or something over them, the bulbs put enough heat out to keep them from getting frost bite and the blanket helps keep the heat in. Maybe something similar with an old pillow case or smaller insulating, think burlap bag, Wrap the water tank with the string of lights and cover with a pillow / burlap bag and use a piece of rope or something to loosely tie it on?
For a bucket of water on the ground, they also make for lizards, heating pads and hot rocks, which are water proof because well, the thing may take a dump on it ..Plug in and it heats up. Get one of those heating a bucket up thats sitting on top of it and it should keep it thawed out. They also make immersion heaters for fish tanks too, that might work to keep the water thawed, and that you would not need to worry about it getting too hot.
Extension cords, yes, get new cords, when they get cold they get brittle, and if the thing is already questionable, you don't need it cracking and exposing wires for you to potentially grab. Since you are not running like a 400 to 1500 watt heater, you don't need to worry about a bunch of amps, so you can get the smaller gauge cords like a 14 or even 16 gauge cord and be fine, and not have to pay 100 dollars for a 12ga or contractor grade cord.
Good luck and hope you can get it worked out. Worrying about the birds is never fun
Aaron