A UPS would help... but it wont last long if you plug it to your PC as PC's power consumption is quite high, depending on your PSU.
Another way is to go for a low power consumption set up like a raspberry pi which is available for sale here on LiveATC and you can contact Dave about it. If you are comfortable with Linux and setting it up then you can refer to aviator's guide here http://milaircomms.com/raspberry_pi_scanner_streaming.html
. It consumes about 5W so your UPS should be able to last longer. Of course you will need to connect the UPS power to your radio too for it to feed to LiveATC.
On another note, what type of lightning protection do you use? Just wondering.