At the large hypermarket in Gallicano they have a not bad selection of electric goods which includes fridges. Only a few kilometers to transport.
Lucca itself has a couple of large shops one of which is called something like ELECTRONICS and one passes it on the left as you travel through Lucca on the way to Barga.
There is also a large MediaWorld which sells fridges, cookers and they deliver to Barga and install, I believe, for about €20.
Here is a link to where it is in Lucca and you can click to see the online shopping and perhaps find something in advance.
Media World - Dove siamo - lucca - Media World